{ "claims": [ { "claim": "With positive, known first-order inclusion probabilities and some zero pairwise inclusion probabilities, the standard Horvitz–Thompson variance estimator has expectation $\\operatorname{Var}(\\hat t)+A$, where $A=\\sum_k\\sum_{l\\ne k:\\,\\pi_{kl}=0}y_ky_l$. Its bias is nonnegative when all outcomes share a sign and can have either sign otherwise.", "claim_id": "P027-H1", "related_corrections": [], "tied_to": "Proposition 1 (printed p. 234, PDF p. 6); abstract; Sections 1, 2.2 and 5" }, { "claim": "The corrected estimator $\\widehat{\\operatorname{Var}}_C(\\hat t)$ adds an estimator of the Young-inequality bound $A^*$ to the Horvitz–Thompson variance estimator. With positive exponents satisfying $1/a_{kl}+1/b_{kl}=1$, its expectation is at least $\\operatorname{Var}(\\hat t)$ for every sampling design with positive first-order inclusion probabilities and every outcome vector, including outcomes of mixed sign.", "claim_id": "P027-H2", "related_corrections": [ "P027-E006", "P027-N002" ], "tied_to": "Proposition 2 (printed pp. 235-236, PDF pp. 7-8); Corollary 2; abstract; Sections 1 and 5" }, { "claim": "The corrected variance estimator is unbiased if, for every pair with $\\pi_{kl}=0$, $|y_k|^{a_{kl}}=|y_l|^{b_{kl}}$ and $y_ky_l\\le0$.", "claim_id": "P027-H3", "related_corrections": [], "tied_to": "Corollary 1 (printed pp. 236-237, PDF pp. 8-9); Section 1 ('no bias under special conditions')" }, { "claim": "The special case $a_{kl}=b_{kl}=2$, denoted $\\widehat{\\operatorname{Var}}_{C2}(\\hat t)$, is conservative in expectation and gives the smallest realized estimator among all choices with constant exponents $(a,b)$.", "claim_id": "P027-H4", "related_corrections": [ "P027-E006" ], "tied_to": "Corollary 2; Corollary 3 (Section 3.2, printed pp. 237-238, PDF pp. 9-10)" }, { "claim": "The bias of the variance estimators is governed by pairs with zero joint inclusion probability. With few such pairs, the paper recommends the Horvitz–Thompson estimator for same-sign outcomes and $\\widehat{\\operatorname{Var}}_C(\\hat t)$ for mixed-sign outcomes; with many such pairs, as in systematic sampling, both can be heavily conservative.", "claim_id": "P027-H5", "related_corrections": [ "P027-E003", "P027-N002" ], "tied_to": "Proposition 1; Proposition 2; Section 4 (printed p. 239, PDF p. 11); abstract; supplement Table A2" }, { "claim": "In one-unit-per-stratum sampling the proposed estimators are less biased than the collapsed-stratum estimator in a range of scenarios.", "claim_id": "P027-H6", "related_corrections": [ "P027-E001", "P027-E006", "P027-A001", "P027-A002" ], "tied_to": "Section 4 (printed p. 240, PDF p. 12); abstract ('may outperform commonly-used approximations'); supplement Section B, Table A1, Figure A1" }, { "claim": "Under systematic PPS sampling with substantial periodicity the proposed estimators perform favourably, while the commonly used with-replacement estimator may be grossly negatively biased.", "claim_id": "P027-H7", "related_corrections": [ "P027-E004", "P027-E007", "P027-N004", "P027-N005" ], "tied_to": "Section 4 (printed p. 240, PDF p. 12); supplement Section C, Table A3, Figure A3" }, { "claim": "Under systematic PPS sampling with a randomly ordered size variable, the paper describes the Horvitz–Thompson variance estimator as conservative but very conservative in magnitude, and reports that the with-replacement estimator performs well.", "claim_id": "P027-H8", "related_corrections": [ "P027-E002", "P027-E003", "P027-E005", "P027-E008" ], "tied_to": "Section 4 (printed p. 239, PDF p. 11); supplement p. A3, Section C, Table A2, Figure A2" }, { "claim": "The proposed variance estimators require no additional assumptions on the outcomes and have no potential for negative bias in expectation. The paper also notes that they may have severe positive bias and perform less well than approximate methods on practical criteria.", "claim_id": "P027-H9", "related_corrections": [], "tied_to": "Proposition 2; Corollary 2; Section 4; Section 5 (printed p. 240, PDF p. 12); Tables A1-A3" } ], "date_published": "2026-10-25", "findings": [ { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Crop of six lines of the supplement. Red boxes mark the words 'estimator dominates' on the third line and 'for smaller stratum sizes and the two perform similarly for larger stratum sizes.' on the fourth line.", "caption": "Supplement, Section B, last paragraph, p. A2 (supplement PDF p. 2). The marked words compare the corrected estimator with the collapsed-stratum estimator by stratum size alone.", "url": "https://pmaronow.github.io/llm-audit/works/P027/exhibits/P027-E001.png" } ], "formal_release": null, "formal_statements": [ { "id": "P027-T14", "url": "https://pmaronow.github.io/llm-audit/works/P027/#P027-T14" }, { "id": "P027-T15", "url": "https://pmaronow.github.io/llm-audit/works/P027/#P027-T15" } ], "headline": "The corrected estimator has smaller averaged bias than the collapsed-stratum estimator exactly when $M<2(1+\\alpha^2)$, and for large strata the two are close only when $\\rho_{ICC}$ is near 1.", "id": "P027-E001", "location": "Supplement, Section B, last paragraph, p. A2 (supplement PDF p. 2); compare Table A1, p. A5", "pdf_page": 2, "printed_page": "A2", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P027-T14", "P027-T15" ], "text": { "as_printed": "``The $(1/N^2)\\widehat{\\operatorname{Var}}_{C2}(\\hat t)$ estimator \\hl{dominates} $\\widehat V_{CS}(\\hat\\mu_{HT})$ \\hl{for smaller stratum sizes and the two perform similarly for larger stratum sizes}.''", "corrected": "``Averaging over the sampling design and the generated populations, with independent unit-variance errors about $\\alpha$ and $-\\alpha$, the $(1/N^2)\\widehat{\\operatorname{Var}}_{C2}(\\hat t)$ estimator has smaller bias than $\\widehat V_{CS}(\\hat\\mu_{HT})$ \\hl{exactly when $M<2(1+\\alpha^2)$}; the two tie when $M=2(1+\\alpha^2)$, and it has larger bias otherwise. This is a comparison of biases, not of mean squared errors. For fixed $\\alpha$ the difference of the expected estimates tends to $1/2$ as $M$ grows and \\hl{their ratio tends to $2-\\rho_{ICC}$}, so large strata alone do not make the two similarly accurate; the expected estimates are relatively close when $\\rho_{ICC}$ is near 1.''", "evidence": "The evidence is a derivation with exact instances. Let $x$ and $z$ be the outcomes of the units drawn from strata 1 and 2. Since $\\pi=1/M$ and $N=2M$, the printed estimators reduce to\n\\[\n\\frac{1}{N^2}\\widehat{\\operatorname{Var}}_{C2}(\\hat t)=\\frac{M-1}{2M}\\left(x^2+z^2\\right),\\qquad \\widehat V_{CS}(\\hat\\mu_{HT})=\\frac14\\left(x-z\\right)^2 .\n\\]\nWith independent unit-variance errors about $\\alpha$ and $-\\alpha$, I have $\\mathbb{E}x^2=\\mathbb{E}z^2=1+\\alpha^2$ and $\\mathbb{E}xz=-\\alpha^2$, and the sampling variance of $\\hat\\mu_{HT}$, averaged over populations, is $(M-1)/(2M)$. Subtracting this variance from the expected estimates, I obtain the averaged biases\n\\[\nB_{C2}=\\frac{M-1}{2M}\\left(1+2\\alpha^2\\right),\\qquad B_{CS}=\\alpha^2+\\frac{1}{2M},\\qquad B_{C2}-B_{CS}=\\frac12-\\frac{1+\\alpha^2}{M}.\n\\]\nBoth biases are nonnegative. The ratio of the two expected estimates is $(1-1/M)(2-\\rho_{ICC})$.\n\n\\begin{audittable}{rrrrrrr}{Exact averaged values under the stated model.}\n$M$ & $\\alpha$ & True variance & Mean of $(1/N^2)\\widehat{\\operatorname{Var}}_{C2}(\\hat t)$ & Mean of $\\widehat V_{CS}$ & $B_{C2}$ & $B_{CS}$ \\\\\n2 & 0 & $1/4$ & $1/2$ & $1/2$ & $1/4$ & $1/4$ \\\\\n5 & 0 & $2/5$ & $4/5$ & $1/2$ & $2/5$ & $1/10$ \\\\\n5 & 2 & $2/5$ & 4 & $9/2$ & $18/5$ & $41/10$ \\\\\n500 & 0 & $499/1000$ & $499/500$ & $1/2$ & $499/1000$ & $1/1000$ \\\\\n\\end{audittable}\n\nEnumerating the 25 samples of a fixed population with $M=5$, I find the same reversal: the biases are $3/10$ and 0 at $\\alpha=0$, and $7/2$ and 4 at $\\alpha=2$.", "problem": "The comparison depends on the stratum means as well as on the stratum size, and the sentence mentions only the size. Averaged over samples and generated populations, the corrected estimator has the smaller bias only when $M<2(1+\\alpha^2)$. At $M=5$ the ranking reverses between $\\alpha=0$ and $\\alpha=2$. For large $M$ the two expected values do not approach each other: their difference tends to $1/2$, which is the limit of the true variance, and their ratio tends to $2-\\rho_{ICC}$. At $M=500$ and $\\alpha=0$ the relative biases are 1 and $1/499$. Table A1 shows the same pattern: its printed nominal bias is larger for $\\widehat{\\operatorname{Var}}_{C2}$ in 23 of the 35 rows, equal in 1 and smaller in 11.", "what_survives": "Table A1, Figure A1, the estimator definitions and the conservativeness of $\\widehat{\\operatorname{Var}}_{C2}$ stand. The first clause holds for the smallest stratum size shown, $M=2$, with a tie at $\\alpha=0$. The main text's weaker statement on p.~240, that these estimators are less biased than the collapsed-stratum estimator ``in a range of scenarios'', also stands. The comparison here is of biases, and nothing is established about mean squared error." }, "url": "https://pmaronow.github.io/llm-audit/works/P027/#P027-E001", "work": { "arxiv_id": null, "doi": null, "title": "Conservative Variance Estimation for Sampling Designs with Zero Pairwise Inclusion Probabilities", "url": "https://pmaronow.github.io/llm-audit/works/P027/" }, "work_id": "P027" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Crop of two lines of text, a displayed formula and two further lines. A red box marks the display 'pi k equals x k over the sum of x l from l = 1 to N, times m'.", "caption": "Supplement, Section C, p. A3 (supplement PDF p. 3). The marked display gives the first order inclusion probability with no bound at 1.", "url": "https://pmaronow.github.io/llm-audit/works/P027/exhibits/P027-E002.png" }, { "alt": "Crop of four lines of the supplement. Red boxes mark the words 'negative binomial distribution with mean 6, variance 16, and offset of 1 to avoid zero values.', which run from the second line into the third.", "caption": "Supplement, Section C, p. A3 (supplement PDF p. 3). The marked words give the law of the sizes, which is unbounded.", "url": "https://pmaronow.github.io/llm-audit/works/P027/exhibits/P027-E002-b.png" } ], "formal_release": null, "formal_statements": [ { "id": "P027-T12", "url": "https://pmaronow.github.io/llm-audit/works/P027/#P027-T12" }, { "id": "P027-T13", "url": "https://pmaronow.github.io/llm-audit/works/P027/#P027-T13" } ], "headline": "The formula $\\pi_k=mx_k/\\sum_lx_l$ can exceed 1 under the stated negative binomial sizes; the recipe needs a rule for such units, for example $\\pi_k=\\min(1,cx_k)$ with $\\sum_k\\pi_k=m$.", "id": "P027-E002", "location": "Supplement, Section C, p. A3 (supplement PDF p. 3): the size variable and the display for $\\pi_k$", "pdf_page": 3, "printed_page": "A3", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P027-T12", "P027-T13" ], "text": { "as_printed": "``For the simulations, a size variable, $x = (x_1, ..., x_N)$, is produced as a vector of $N$ random draws from a \\hl{negative binomial distribution with mean 6, variance 16, and offset of 1} to avoid zero values.'' Later on the same page: ``Systematic PPS sampling was implemented such that each unit's first order inclusion probability is given by,\n\\[\n\\hl{\\pi_k = \\frac{x_k}{\\sum_{l=1}^{N} x_l}\\, m,}\n\\]\nwhere $m$ is the size of the systematic sample, and joint inclusion probabilities are computed as per Madow (1949). For the simulations, sampling and computation of inclusion probabilities were carried out using functions provided by Tillé and Matei (2011).''", "corrected": "One possible repair is:\n``Systematic PPS sampling can use first order inclusion probabilities given by,\n\\[\n\\pi_k = \\hl{\\min\\left(1,\\,c\\,x_k\\right),\\qquad c \\text{ chosen so that } \\sum_{k=1}^{N}\\pi_k=m,}\n\\]\nwhere $m$ is the size of the systematic sample\\hl{; a systematic sample is drawn with these probabilities in the given order of the population,} and joint inclusion probabilities are \\hl{those of that design}.'' Alternatively the printed display is kept and the size vector is redrawn until $m\\max_kx_k\\le\\sum_{l=1}^{N}x_l$, conditioning the size-generation law on this inequality.\n\nWhen no unit is oversized, $c=m/\\sum_lx_l$ and the display reduces to the printed one. On the example above, $c=1/7$ gives $(\\pi_1,\\dots,\\pi_8)=(1,1/7,\\dots,1/7)$, which is realized by the seven samples $\\{1,j\\}$, $j=2,\\dots,8$, each with probability $1/7$. Capping the printed values at 1 without redistributing gives a total of $51/37$, not 2. The actual implementation is needed to establish whether either rule matches the reported runs.", "evidence": "The evidence is an exact counterexample and an exact calculation. For $N=8$, $m=2$ and $x=(30,1,\\dots,1)$ I have $\\sum_lx_l=37$ and $(\\pi_1,\\dots,\\pi_8)=(60/37,2/37,\\dots,2/37)$. The entries sum to 2, but $\\pi_1>1$. Note that $\\mathbb{E}(I_1)\\le 1$ for any sampling design. A negative binomial law with mean 6 and variance 16 before the offset has success probability $3/8$ and size parameter $18/5$, and it assigns positive probability to every nonnegative integer. The shifted law therefore gives the vector above positive probability. With independent draws under this interpretation, the probability that some $\\pi_k$ exceeds 1 lies between 0.000168504 and 0.000168505. Each of the six pairs $(N,m)$ of Table A2 admits such a vector. Since the simulation code was unavailable to me, I do not know whether such a case arose or how it was handled.", "problem": "Since the sizes are unbounded, $m\\max_kx_k$ can exceed $\\sum_lx_l$. The display then gives a value above 1, which is not the expectation of an inclusion indicator. In the scenario $N=8$, $m=2$ the size vector $(30,1,1,1,1,1,1,1)$ has $\\pi_1=60/37$. Under the stated law of the sizes some $\\pi_k$ exceeds 1 with probability about 0.00017 in that scenario. The supplement states no rule for such units, and the simulation cannot be reproduced from its description when they occur.", "what_survives": "No entry of Table A2 and no comparison is shown to be wrong. The results of the main text start from valid inclusion probabilities and are untouched. Section B is also untouched, as is the periodic design of Table A3, where $\\max_k\\pi_k=10m/(6N)\\le 5/12$." }, "url": "https://pmaronow.github.io/llm-audit/works/P027/#P027-E002", "work": { "arxiv_id": null, "doi": null, "title": "Conservative Variance Estimation for Sampling Designs with Zero Pairwise Inclusion Probabilities", "url": "https://pmaronow.github.io/llm-audit/works/P027/" }, "work_id": "P027" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Crop of five lines of the supplement. A red box marks the words 'y is proportional to x' in the sentence that begins 'Because y is proportional to x with x greater than or equal to 1, Proposition 1 in the main text indicates'.", "caption": "Supplement, Section C, p. A3 (supplement PDF p. 3). The marked premise is that $y$ is proportional to $x$.", "url": "https://pmaronow.github.io/llm-audit/works/P027/exhibits/P027-E003.png" }, { "alt": "Crop of two lines of text, a displayed formula and two further lines. A red box marks the display 'y k equals beta x k plus epsilon k'. The line below it says that epsilon k is Gaussian noise with mean zero.", "caption": "Supplement, Section C, p. A3 (supplement PDF p. 3). The marked display is the outcome model, in which Gaussian noise is added to $\\beta x_k$.", "url": "https://pmaronow.github.io/llm-audit/works/P027/exhibits/P027-E003-b.png" } ], "formal_release": null, "formal_statements": [ { "id": "P027-T10", "url": "https://pmaronow.github.io/llm-audit/works/P027/#P027-T10" }, { "id": "P027-T11", "url": "https://pmaronow.github.io/llm-audit/works/P027/#P027-T11" } ], "headline": "The simulated outcomes $y_k=\\beta x_k+\\epsilon_k$ are not proportional to $x$ and can change sign; Proposition 1 then implies that the uncorrected estimator is conservative only on average over the generated outcomes, not for each generated population.", "id": "P027-E003", "location": "Supplement, Section C, p. A3 (supplement PDF p. 3), paragraph beginning “Because $y$ is proportional to $x$”", "pdf_page": 3, "printed_page": "A3", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P027-T10", "P027-T11" ], "text": { "as_printed": "``Because \\hl{$y$ is proportional to $x$} with $x \\geq 1$, Proposition 1 in the main text indicates that the Horvitz-Thompson variance estimator for $\\hat\\mu_{HT}$, $(1/N^2)\\widehat{\\operatorname{Var}}(\\hat t)$, provides a conservative approximation of the sampling variance of $\\hat\\mu_{HT}$.''\n\nThe outcome model is displayed earlier on the same page: ``$y_k = \\beta x_k + \\epsilon_k$,'' where ``$\\epsilon_k$ is Gaussian noise with mean zero and standard deviation $\\sigma_y\\sqrt{1-\\rho^2}$'' and ``we fix $\\sigma_y = 30$''.", "corrected": "``For a size vector $x$ giving \\hl{valid inclusion probabilities}, \\hl{assuming independent mean-zero errors conditional on $x$}, the model has \\hl{$\\mathrm{E}_\\epsilon[y_k\\mid x]=\\beta x_k$} with $x_k\\geq 1$. Proposition 1 in the main text therefore indicates that the Horvitz-Thompson variance estimator for $\\hat\\mu_{HT}$, $(1/N^2)\\widehat{\\operatorname{Var}}(\\hat t)$, has nonnegative bias \\hl{after averaging over the generated outcomes}:\n\\[\n\\mathrm{E}_\\epsilon[A\\mid x]=\\beta^2\\sum_{k\\ne l:\\,\\pi_{kl}=0}x_kx_l\\geq 0.\n\\]\nIt \\hl{does not guarantee} a conservative approximation for each generated finite population, because positive sizes and Gaussian errors do not ensure outcomes of one sign.''", "evidence": "The evidence is an exact counterexample and a derivation. I set $N=8$ and $m=2$, with sizes and outcomes\n\\[\nx=(1,3,1,3,1,3,1,3),\\qquad y=(1,-1,1,-1,1,-1,1,-1).\n\\]\nThen $\\pi_k=x_k/8$, and systematic PPS sampling with a uniform start on $[0,8)$ has four possible samples.\n\n\\begin{audittable}{llrrr}{The eight-unit example, enumerated in full.}\nStart & Sample & Probability & $\\hat t$ & $\\widehat{\\operatorname{Var}}(\\hat t)$ \\\\\n$[0,1)$ & $\\{1,5\\}$ & $1/8$ & 16 & 224 \\\\\n$[1,4)$ & $\\{2,6\\}$ & $3/8$ & $-16/3$ & $160/9$ \\\\\n$[4,5)$ & $\\{3,7\\}$ & $1/8$ & 16 & 224 \\\\\n$[5,8)$ & $\\{4,8\\}$ & $3/8$ & $-16/3$ & $160/9$ \\\\\n\\end{audittable}\n\nFrom the table, I find $\\operatorname{Var}(\\hat t)=256/3$ and $\\mathbb{E}\\,\\widehat{\\operatorname{Var}}(\\hat t)=208/3$, which implies that $A=-16$. For the mean, the sampling variance is $4/3$ and the expected estimate is $13/12$, a bias of $-1/4$. Every outcome vector within $1/8$ of this $y$ in each coordinate has $A\\le -13/4$, and this set has positive probability under independent Gaussian noise for every listed $\\rho<1$.\n\nFor the average, fix a feasible size vector $x$ and suppose that the errors are independent with mean zero. For $k\\ne l$ I have $\\mathbb{E}_\\epsilon[y_ky_l\\mid x]=\\beta^2x_kx_l$. Summing over the pairs that are never selected together, I obtain\n\\[\n\\mathbb{E}_\\epsilon[A\\mid x]=\\beta^2\\sum_{k\\ne l:\\,\\pi_{kl}=0}x_kx_l\\ge 0 .\n\\]", "problem": "By Proposition 1, the bias is nonnegative when all outcomes have the same sign. The outcomes of this simulation are $\\beta x_k$ plus Gaussian noise with standard deviation $30\\sqrt{1-\\rho^2}$. They are not proportional to $x$ and need not share a sign, and the bias term $A$ can be negative for a generated population. A weaker guarantee does hold: with independent errors the bias is nonnegative after averaging over the generated outcomes.", "what_survives": "Proposition 1, the same-sign condition as the main text states it on p.~234, and the conservativeness of the corrected estimator stand; the bias of $\\widehat{\\operatorname{Var}}_{C2}(\\hat t)$ is $(1/2)\\sum_{k\\ne l:\\,\\pi_{kl}=0}(y_k+y_l)^2\\ge 0$. The positive averaged biases of Table A2 agree with the averaged statement and are not in question. Only the guarantee for each generated population of this simulation is lost, and the label ``Conservative (HT)'' of Figure A2 is to be read as a statement about the average over outcomes." }, "url": "https://pmaronow.github.io/llm-audit/works/P027/#P027-E003", "work": { "arxiv_id": null, "doi": null, "title": "Conservative Variance Estimation for Sampling Designs with Zero Pairwise Inclusion Probabilities", "url": "https://pmaronow.github.io/llm-audit/works/P027/" }, "work_id": "P027" }, { "before_review": { "class": "error", "severity": "Moderate" }, "class": "clarity_note", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Crop of the last four lines of the text of the supplement. Red boxes mark the words 'almost perfectly.' on the second line, 'canceling each other out,' at the end of the third line and 'in which case A tends to 0.' on the fourth line.", "caption": "Supplement, Section C, last paragraph before the References, p. A4 (supplement PDF p. 4). The marked words are the accuracy claim and the explanation given for it, both of which need qualifying.", "url": "https://pmaronow.github.io/llm-audit/works/P027/exhibits/P027-E004.png" }, { "alt": "The title reads 'Table A3: printed relative biases and the derived ratio of outcome averages'. Two panels, one for sampling fraction one quarter and one for one twentieth, plot relative bias from minus 0.175 to 0.03 against rho from 0 to 1. Blue open markers joined by lines are the printed values for three designs in each panel. In the left panel they lie between minus 0.14 and minus 0.09, near an orange dashed line at minus 5 over 36, about minus 0.139. In the right panel they lie between minus 0.03 and minus 0.01, near an orange dashed line at minus 5 over 180, about minus 0.028. A grey line at zero is labelled as a reference line for zero relative bias. No printed value is at or above zero.", "caption": "Relative bias of the uncorrected estimator in the 60 rows of supplement Table A3, as printed there to two decimals, with the ratio $-5/(9q)$ derived for the stated design (dashed) and zero as a reference line. The derived ratio is a ratio of averages over outcomes and is not a recomputation of the table.", "url": "https://pmaronow.github.io/llm-audit/works/P027/exhibits/P027-E004-ht-relative-bias-r2.png" } ], "formal_release": null, "formal_statements": [ { "id": "P027-T1", "url": "https://pmaronow.github.io/llm-audit/works/P027/#P027-T1" }, { "id": "P027-T2", "url": "https://pmaronow.github.io/llm-audit/works/P027/#P027-T2" } ], "headline": "In the periodic design the elements of the bias term cancel only in part: $A=-\\sum_gT_g^2$, minus the sum of the squared outcome totals of the $q=N/m$ possible samples, so the uncorrected estimator is biased downward unless every such total is zero; under the stated outcome model the ratio of bias averaged over outcomes to variance averaged over outcomes is $-5/(9q)$.", "id": "P027-E004", "location": "Supplement, Section C, last paragraph before the References, p. A4 (supplement PDF p. 4); compare Table A3, p. A8", "pdf_page": 4, "printed_page": "A4", "reaches": [], "related_entries": [], "revisions": [ { "classification_after": { "class": "clarity_note", "severity": null }, "classification_before": { "class": "error", "severity": "Moderate" }, "note": "Reclassified from a Moderate error to a clarity note. The derivation shows partial cancellation in $A$, but does not establish that the paragraph’s qualitative description is false. The paragraph gives no numerical threshold for “almost perfectly” or scale or limit for “$A$ tends to 0”, and Table A3 prints the negative bias. The derivation, exact example and figure are retained." } ], "same_defect_as": [], "severity": null, "subject": "paper", "target_ids": [ "P027-T1", "P027-T2" ], "text": { "as_printed": "``Interestingly, $\\widehat{\\operatorname{Var}}(\\hat t)$ tends to match the true value of the variance \\hl{almost perfectly}. This occurs because of the special nature of the data generating process, which results in the elements in the bias term, $A$, \\hl{canceling each other out, in which case $A$ tends to 0}.''", "corrected": "A clarified version of the two sentences is:\n\n``Interestingly, $\\widehat{\\operatorname{Var}}(\\hat t)$ tends to be \\hl{close to} the true value of the variance, \\hl{especially at the smaller sampling fraction}. This occurs because of the special nature of the data generating process, which results in the elements in the bias term, $A$, \\hl{partly canceling each other out}. \\hl{The cancellation is not complete}: if $T_g$ is the demeaned outcome total of the $g$th of the $q=N/m$ possible samples, then\n\\[\n\\hl{A=-\\sum_{g=1}^{q}T_g^2\\le 0,}\n\\]\n\\hl{so that $\\widehat{\\operatorname{Var}}(\\hat t)$ is biased downward unless every $T_g$ is zero}. Under the outcome model used here, the ratio of the bias averaged over outcomes to the variance averaged over outcomes is \\hl{$-5/(9q)$}, about $-3\\%$ when $m/N=1/20$ and $-14\\%$ when $m/N=1/4$.''", "intended_meaning": "The paragraph reports the simulation results of Table A3 and Figure A3 and offers a qualitative reason for them. It gives ``almost perfectly'' no numerical threshold, and it gives ``$A$ tends to 0'' no scale and no limiting operation. On its natural reading it describes approximate agreement explained by approximate cancellation, and on that reading it is not false. The cancellation is partial, however, and what it leaves is a systematic downward bias, which the paragraph does not mention; Table A3 itself prints that bias.\n\nIn the six designs of Table A3 the possible systematic samples are $q=N/m$ disjoint groups of units, with $q=4$ or $q=20$. With demeaned outcomes the products that make up $A$ can have both signs and offset one another, but their sum equals minus the sum of the squared group totals. It is zero only when every group total is zero, and demeaning makes the totals sum to zero without forcing each of them to be zero. The uncorrected estimator therefore has nonpositive bias in this design. Under the stated outcome model the ratio of the averaged bias to the averaged variance is $-5/(9q)$, about $-14\\%$ at $m/N=1/4$ and $-3\\%$ at $m/N=1/20$. Table A3 prints relative biases between $-0.03$ and $-0.01$ for $q=20$ and between $-0.14$ and $-0.09$ for $q=4$. The agreement is therefore close at the smaller sampling fraction, while at the larger one the printed shortfall is 9 to 14 per cent of the true variance. Taken literally, as a statement about the unscaled term, ``$A$ tends to 0'' is misleading: for $\\rho>0$ and a fixed sampling fraction, $A/N^2$ does not vanish as $N$ grows. At a fixed sample size, by contrast, the ratio $-5/(9q)$ tends to zero as $N$ grows. The paragraph specifies neither limit.\n\nThe evidence is a derivation and an exact example. Let $z_k=\\tilde y_k$ be the demeaned outcomes. Since the sizes alternate between 2 and 10 and $q$ is even, a systematic PPS draw selects one of the groups $G_g=\\{g,g+q,\\dots,g+(m-1)q\\}$, $g=1,\\dots,q$, with probability $p_g=1/(3q)$ for odd $g$ and $p_g=5/(3q)$ for even $g$. Two distinct units have $\\pi_{kl}=0$ exactly when they lie in different groups. Let $T_g=\\sum_{k\\in G_g}z_k$, and note that $\\sum_gT_g=0$ because the outcomes are demeaned. By the definition of $A$ on p.~234 of the main text, I have\n\\[\nA=\\sum_{g}\\sum_{h\\ne g}T_gT_h=\\Bigl(\\sum_gT_g\\Bigr)^2-\\sum_gT_g^2=-\\sum_gT_g^2\\le 0,\n\\]\nand the bias of $(1/N^2)\\widehat{\\operatorname{Var}}(\\hat t)$ is $A/N^2$.\n\nFor an exact instance, let $N=8$, $m=2$, with sizes $(2,10,2,10,2,10,2,10)$, demeaned outcomes $(-1,1,-1,1,-1,1,-1,1)$ and a uniform start on $[0,24)$.\n\n\\begin{audittable}{llrrrrr}{The eight-unit periodic example, enumerated in full. The true variance of $\\hat\\mu_{HT}$ is $9/5$.}\nStart & Sample & Probability & $T_g$ & $\\hat\\mu_{HT}$ & $(1/N^2)\\widehat{\\operatorname{Var}}(\\hat t)$ & $(1/N^2)\\widehat{\\operatorname{Var}}_{C2}(\\hat t)$ \\\\\n$[0,2)$ & $\\{1,5\\}$ & $1/12$ & $-2$ & $-3$ & $33/4$ & $21/2$ \\\\\n$[2,12)$ & $\\{2,6\\}$ & $5/12$ & 2 & $3/5$ & $21/100$ & $33/50$ \\\\\n$[12,14)$ & $\\{3,7\\}$ & $1/12$ & $-2$ & $-3$ & $33/4$ & $21/2$ \\\\\n$[14,24)$ & $\\{4,8\\}$ & $5/12$ & 2 & $3/5$ & $21/100$ & $33/50$ \\\\\nExpectation & & & & 0 & $31/20$ & $23/10$ \\\\\n\\end{audittable}\n\nHere $A=-16$, the bias is $A/N^2=-1/4$, and the relative bias is $-5/36$. The 48 products $z_kz_l$ over ordered pairs of units in different groups are each $\\pm1$: 16 are positive and 32 negative, so that the absolute value of their sum is a third of the sum of their absolute values. With the same alternating outcomes, the bias is $-1/4$ in each of the three designs with $q=4$. In each of the three with $q=20$ it is $-1/20$, a relative bias of $-1/36$, and the absolute value of the sum of the products is a nineteenth of the sum of their absolute values.\n\nUnder independent Gaussian residuals with a common variance, $\\mathbb{E}_y(T_g^2)$ takes the same value $D$ for every group. Then I have $\\mathbb{E}_y(A)=-qD$ and $\\mathbb{E}_y\\operatorname{Var}(\\hat t)=D\\sum_g1/p_g=9q^2D/5$, and their ratio is $-5/(9q)$. This is a ratio of averages over outcomes. It is not generally the relative bias of an individual population: when its variance is positive, that bias lies between $-5/(3q)$ and $-1/(3q)$. It does not reconstruct the entries of Table A3. With the sample standard deviation of $x$ in $\\beta$, the quantity $A/N^2$ converges in probability to $-900\\rho^2/q$ as $N$ grows with $q$ and $\\rho>0$ fixed. Table A3 prints a negative bias for the uncorrected estimator in all 60 rows.\n\nProposition 1, the corrected estimator and its conservativeness, and the main text's comparison with the with-replacement estimator under periodicity are unaffected. In the exact example the corrected estimator has expectation $23/10$ against a true variance of $9/5$, and the with-replacement estimator is identically zero. The earlier sentence on p.~A4, that $\\widehat{\\operatorname{Var}}(\\hat t)$ ``may have either positive or negative bias'', is correct as a general statement, although in this design the bias is never positive. This note does not revise the entries of Table A3; \\fid{P027-N004} concerns four of its rows, and \\fid{P027-E007} the label of the uncorrected series." }, "url": "https://pmaronow.github.io/llm-audit/works/P027/#P027-E004", "work": { "arxiv_id": null, "doi": null, "title": "Conservative Variance Estimation for Sampling Designs with Zero Pairwise Inclusion Probabilities", "url": "https://pmaronow.github.io/llm-audit/works/P027/" }, "work_id": "P027" }, { "before_review": null, "class": "clarity_note", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "complete_derivation", "exhibits": [ { "alt": "Crop of the display 'y k equals beta x k plus epsilon k' and the six lines below it. Red boxes mark the words 'sigma x and sigma y are the standard deviation of the N values in x and y respectively, where E x of sigma x = 4 and we fix sigma y = 30', which run over two lines.", "caption": "Supplement, Section C, p. A3 (supplement PDF p. 3). The marked words define $\\sigma_x$ and $\\sigma_y$ as standard deviations of the realized values, set $\\mathrm{E}_x(\\sigma_x)=4$ and fix $\\sigma_y=30$.", "url": "https://pmaronow.github.io/llm-audit/works/P027/exhibits/P027-E005.png" } ], "formal_release": null, "formal_statements": [], "headline": "The values 4, 30 and $\\rho$ are nominal settings of the simulation; the standard deviations and, when defined, the correlation of the $N$ realized values vary from one generated population to the next.", "id": "P027-E005", "location": "Supplement, Section C, p. A3 (supplement PDF p. 3), the definitions after the display $y_k=\\beta x_k+\\epsilon_k$", "pdf_page": 3, "printed_page": "A3", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": null, "subject": "paper", "target_ids": [], "text": { "as_printed": "``for $k = 1, ..., N$, where $\\beta = \\rho(\\sigma_y/\\sigma_x)$, $\\epsilon_k$ is Gaussian noise with mean zero and standard deviation $\\sigma_y\\sqrt{1-\\rho^2}$, and \\hl{$\\sigma_x$ and $\\sigma_y$ are the standard deviation of the $N$ values in $x$ and $y$ respectively, where $\\mathrm{E}_x(\\sigma_x) = 4$ and we fix $\\sigma_y = 30$} (again, to mimic data distributed as in Wolter (2007, Ch. 8)). Thus, $x$ and $y$ have a linear relationship through the origin and \\hl{their correlation is given by $\\rho$}.''", "corrected": "``for $k = 1, ..., N$, where $\\beta = \\rho(\\sigma_y/\\sigma_x)$, $\\epsilon_k$ is Gaussian noise with mean zero and standard deviation $\\sigma_y\\sqrt{1-\\rho^2}$, $\\sigma_y = 30$ is a \\hl{nominal} outcome scale and $\\rho$ is a \\hl{nominal} correlation. The size distribution has standard deviation 4. The standard deviations and, \\hl{when defined}, the correlation of the $N$ realized values \\hl{vary across generated populations}.'' The supplement should also state which standard deviation of the sizes enters $\\beta$, the realized one or the value 4, and how a realized standard deviation of zero is handled. The simulation code or an explicit description is needed to settle this.", "intended_meaning": "The paragraph sets the nominal calibration of the simulation: sizes drawn from a law with standard deviation 4, an outcome scale of 30 and a correlation parameter $\\rho$. Read literally as statements about the $N$ realized values, they do not hold. The expected standard deviation of a finite sample from a law with variance 16 is below 4 by Jensen's inequality, and with Gaussian noise the realized standard deviation of $y$ and, when defined, the realized correlation of $x$ and $y$ are random. The nominal reading is coherent, and no entry of Tables A2 and A3 is shown to depend on the difference. The supplement does not say whether $\\beta$ uses the realized standard deviation of the sizes or the value 4, nor what is done when the realized standard deviation is zero." }, "url": "https://pmaronow.github.io/llm-audit/works/P027/#P027-E005", "work": { "arxiv_id": null, "doi": null, "title": "Conservative Variance Estimation for Sampling Designs with Zero Pairwise Inclusion Probabilities", "url": "https://pmaronow.github.io/llm-audit/works/P027/" }, "work_id": "P027" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "complete_derivation", "exhibits": [ { "alt": "Crop of the displayed inequality of Corollary 2 and the one-line proof beneath it. A red box marks the words 'Proposition 1' in the sentence 'Proposition 1 therefore holds.'", "caption": "Proof of Corollary 2, p. 237 (PDF p. 9). The marked reference should be to Proposition 2.", "url": "https://pmaronow.github.io/llm-audit/works/P027/exhibits/P027-E006.png" }, { "alt": "Crop of five lines of the first paragraph of the introduction. A red box marks the word 'remains' in 'Optimal methods for variance estimation in these cases remains an open problem.'", "caption": "Section 1, first paragraph, p. 231 (PDF p. 3). The marked verb should be “remain”.", "url": "https://pmaronow.github.io/llm-audit/works/P027/exhibits/P027-E006-b.png" }, { "alt": "Crop of the end of page A1 of the supplement, with two displayed formulas. A red box marks the case '0 if s = t' in the display for the joint inclusion probability, and a second red box marks the symbol epsilon with subscript 'is' in the display for the outcome.", "caption": "Supplement, Section B, foot of p. A1 (supplement PDF p. 1). The marked case lacks the restriction to distinct units, and the marked residual should carry the index $k$.", "url": "https://pmaronow.github.io/llm-audit/works/P027/exhibits/P027-E006-c.png" } ], "formal_release": null, "formal_statements": [ { "id": "P027-T18", "url": "https://pmaronow.github.io/llm-audit/works/P027/#P027-T18" }, { "id": "P027-T23", "url": "https://pmaronow.github.io/llm-audit/works/P027/#P027-T23" } ], "headline": "Four misprints: the proof of Corollary 2 should cite Proposition 2, “remains” should read “remain”, the display for $\\pi_{ks,lt}$ in the supplement applies to distinct units only, and $\\epsilon_{is}$ should read $\\epsilon_{ks}$.", "id": "P027-E006", "location": "Proof of Corollary 2, p. 237 (PDF p. 9); Section 1, first paragraph, p. 231 (PDF p. 3); supplement, Section B, pp. A1–A2 (supplement PDF pp. 1–2)", "pdf_page": 9, "printed_page": "237", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P027-T18", "P027-T23" ], "text": { "as_printed": "\\begin{itemize}\n\\item (a) p.~237: ``Proof. For all pairs $k, l$ such that $\\pi_{kl} = 0$, $1/a_{kl} + 1/b_{kl} = 1$. \\hl{Proposition 1} therefore holds.''\n\\item (b) p.~231: ``Optimal methods for variance estimation in these cases \\hl{remains} an open problem.''\n\\item (c) Supplement, p.~A1: ``and for units $k$ and $l$ in strata 1 or 2, we have'' followed by the display\n\\[\n\\pi_{ks,lt} = \\begin{cases} \\hl{0} & \\hl{\\text{if } s = t} \\\\ \\frac{1}{M^2} & \\text{if } s \\neq t \\end{cases}.\n\\]\n\\item (d) Supplement, pp.~A1--A2: ``In stratum $s$, outcomes $y_s = (y_{1s}, ..., y_{Ms})$ are drawn as, $y_{ks} = \\alpha_s + \\hl{\\epsilon_{is}}$, where $\\alpha_s$ is a stratum-specific constant and $\\epsilon_{ks} \\sim \\mathrm{N}(0, 1)$.''\n\\end{itemize}", "corrected": "\\begin{itemize}\n\\item (a) ``Proof. For all pairs $k, l$ such that $\\pi_{kl} = 0$, $1/a_{kl} + 1/b_{kl} = 1$. \\hl{Proposition 2 therefore applies.}''\n\\item (b) ``Optimal methods for variance estimation in these cases \\hl{remain} an open problem.''\n\\item (c) ``and for \\hl{distinct} units \\hl{$(k,s)\\neq(l,t)$} in strata 1 or 2, we have'' followed by the same display, with $\\pi_{ks,ks}=1/M$ for a unit paired with itself.\n\\item (d) ``$y_{ks} = \\alpha_s + \\hl{\\epsilon_{ks}}$''\n\\end{itemize}", "evidence": "This correction rests on a check of the pages. Proposition 1 (p.~234) is the bias identity, and Proposition 2 (p.~235) is the inequality $\\mathbb{E}[\\widehat{\\operatorname{Var}}_C(\\hat t)]\\ge\\operatorname{Var}(\\hat t)$ under $1/a_{kl}+1/b_{kl}=1$. Since the main text sets $\\pi_{kk}=\\pi_k$ on p.~233, I have $\\pi_{ks,ks}=\\pi_{ks}=1/M$. Page A2 continues with $\\epsilon_{ks}$.", "problem": "I identify four local slips. In (a) the inequality of Corollary 2 follows from Proposition 2, whose hypothesis the proof checks. In (c) the first case, taken literally for a unit paired with itself, gives $\\pi_{ks,ks}=0$ in place of $1/M$. In (d) the index $i$ is not defined.", "what_survives": "These slips do not affect Corollary 2 or the simulation design." }, "url": "https://pmaronow.github.io/llm-audit/works/P027/#P027-E006", "work": { "arxiv_id": null, "doi": null, "title": "Conservative Variance Estimation for Sampling Designs with Zero Pairwise Inclusion Probabilities", "url": "https://pmaronow.github.io/llm-audit/works/P027/" }, "work_id": "P027" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "recalculation_from_source", "exhibits": [ { "alt": "Crop of the caption of Figure A3 and its first two panels, for N = 8, m = 2 and for N = 40, m = 2, each plotting E(V) against rho. A red box marks the legend entry 'Conservative (HT)' in the first panel. In both panels the short-dashed curve of open circles that this entry names lies below the solid curve of filled circles labelled 'True'.", "caption": "Supplement, Figure A3, p. A10 (supplement PDF p. 10), caption and first row of panels. The marked legend entry calls the uncorrected series conservative; its curve, the short-dashed line of open circles, lies below the solid “True” curve.", "url": "https://pmaronow.github.io/llm-audit/works/P027/exhibits/P027-E007.png" }, { "alt": "Crop of four lines of the supplement. Red boxes mark the label 'Conservative (HT)', which is broken across the second and third lines as 'Con-' and 'servative (HT)'.", "caption": "Supplement, Section C, p. A4 (supplement PDF p. 4). The marked words give the same label in the text.", "url": "https://pmaronow.github.io/llm-audit/works/P027/exhibits/P027-E007-b.png" } ], "formal_release": null, "formal_statements": [], "headline": "In the periodic simulation the uncorrected Horvitz–Thompson series is not conservative; it should be labelled “Uncorrected HT”, not “Conservative (HT)”.", "id": "P027-E007", "location": "Supplement, Figure A3, legend of the first panel, p. A10 (supplement PDF p. 10); Section C, p. A4 (supplement PDF p. 4), paragraph beginning “Table A3 and Figure A3 display”", "pdf_page": 10, "printed_page": "A10", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "Supplement, p.~A10, legend of Figure A3: ``\\hl{Conservative} (HT)''.\n\nSupplement, p.~A4: ``Each graph plots the true variance (Var or ``True''), expected value of $(1/N^2)\\widehat{\\operatorname{Var}}(\\hat t)$ ($\\widehat{\\operatorname{Var}}_{HT}$ or ``\\hl{Conservative} (HT)''), expected value of $(1/N^2)\\widehat{\\operatorname{Var}}_{C2}(\\hat t)$ ($\\widehat{\\operatorname{Var}}_{C2}$ or ``Conservative ($A^*$)''), and expected value of $\\widehat{\\operatorname{Var}}_{WR}(\\hat\\mu_{HT})$ (``With-repl.'') over values of $\\rho$ and for varying sample sizes ($m$) and sampling proportions ($m/N$).''", "corrected": "Legend of Figure A3: ``\\hl{Uncorrected HT}''.\n\nSupplement, p.~A4: ``\\ldots expected value of $(1/N^2)\\widehat{\\operatorname{Var}}(\\hat t)$ ($\\widehat{\\operatorname{Var}}_{HT}$ or ``\\hl{Uncorrected HT}''), \\ldots''", "evidence": "This is a recalculation from Table A3. The printed mean of $\\widehat{\\operatorname{Var}}_{HT}$ is below the printed true variance in each of the 60 rows, and each printed nominal bias is negative. In row 10, for example, $1227.24-1397.05=-169.81$.", "problem": "The label calls the uncorrected estimator conservative, but in this simulation its mean is below the true variance in every row of Table A3. The curve is identified by its formula, and the same page says that the estimator may have bias of either sign. Only the descriptor is wrong. Figure A2 uses the same descriptor.", "what_survives": "The estimator formulas, the tabulated values and the comparisons are unaffected. The sign of the bias in this design is derived in \\fid{P027-E004}." }, "url": "https://pmaronow.github.io/llm-audit/works/P027/#P027-E007", "work": { "arxiv_id": null, "doi": null, "title": "Conservative Variance Estimation for Sampling Designs with Zero Pairwise Inclusion Probabilities", "url": "https://pmaronow.github.io/llm-audit/works/P027/" }, "work_id": "P027" }, { "before_review": null, "class": "clarity_note", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Crop of the caption of Figure A2 and its first two panels, for N = 8, m = 2 and for N = 40, m = 2, each plotting E(V) from 0 to 2000 against rho. Red boxes mark the upper right corner of each panel, where the dashed curve of open circles labelled 'Conservative (HT)' rises through the top edge of the frame.", "caption": "Supplement, Figure A2, p. A9 (supplement PDF p. 9), caption and first row of panels. The marked corners are where the dashed Horvitz–Thompson curve leaves the frame; the markers at $\\rho=0.99$ for $N=8$ and at $\\rho=0.95$ and $0.99$ for $N=40$ are not drawn. The same happens at $\\rho=0.99$ in the panels for $N=600$ and $N=1200$, which are not shown here.", "url": "https://pmaronow.github.io/llm-audit/works/P027/exhibits/P027-E008.png" } ], "formal_release": null, "formal_statements": [], "headline": "Five Horvitz–Thompson markers at $\\rho=0.95$ and $\\rho=0.99$ lie above the plotting range of Figure A2 and are not drawn; Table A2 gives their values.", "id": "P027-E008", "location": "Supplement, Figure A2, p. A9 (supplement PDF p. 9); the values are in Table A2, p. A7", "pdf_page": 9, "printed_page": "A9", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": null, "subject": "paper", "target_ids": [], "text": { "as_printed": "``Figure A2: Expected value of estimators of sampling variance of $\\hat\\mu_{HT}$ under PPS systematic sampling for different values of $\\rho$, sample sizes ($m$), and sampling proportions ($m/N$): random size case.'' The vertical axes of the six panels are labelled from 0 to 2000, and the plotting area ends a little above 2000. The dashed curve labelled ``Conservative (HT)'' \\hl{leaves the top of the frame} in the panels for $(N,m)=(8,2)$, $(40,2)$, $(600,30)$ and $(1200,60)$, with no marker or note for the points that fall outside.", "corrected": "The vertical range of every panel is extended to include 2667.59, or a note is added under the figure: ``The Horvitz-Thompson values 2236.50 ($N=8$, $\\rho=0.99$), 2473.52 and 2667.59 ($N=40$, $\\rho=0.95$ and $0.99$), 2383.53 ($N=600$, $\\rho=0.99$) and 2350.79 ($N=1200$, $\\rho=0.99$) \\hl{lie above the plotting range}; see Table A2.''", "intended_meaning": "Nothing printed is false. The curves visibly run out of the frame, and Table A2 lists every value. The five markers that are not drawn have the values 2236.50 ($N=8$, $m=2$, $\\rho=0.99$), 2473.52 and 2667.59 ($N=40$, $m=2$, $\\rho=0.95$ and $0.99$), 2383.53 ($N=600$, $m=30$, $\\rho=0.99$) and 2350.79 ($N=1200$, $m=60$, $\\rho=0.99$). The markers at $\\rho=0.95$ for $N=600$ (2173.08) and $N=1200$ (2171.46) exceed 2000 and remain visible." }, "url": "https://pmaronow.github.io/llm-audit/works/P027/#P027-E008", "work": { "arxiv_id": null, "doi": null, "title": "Conservative Variance Estimation for Sampling Designs with Zero Pairwise Inclusion Probabilities", "url": "https://pmaronow.github.io/llm-audit/works/P027/" }, "work_id": "P027" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Crop of the heading 'References' and the two-line entry for Hansen, Hurwitz and Madow (1953). A red box marks the words 'Methods and Theory.' at the start of the second line of the entry.", "caption": "First entry of the References, p. 240 (PDF p. 12). The marked subtitle should read “Methods and Applications”.", "url": "https://pmaronow.github.io/llm-audit/works/P027/exhibits/P027-E009.png" } ], "formal_release": null, "formal_statements": [], "headline": "Volume I of Hansen, Hurwitz and Madow (1953) is subtitled “Methods and Applications”, not “Methods and Theory”.", "id": "P027-E009", "location": "References, entry for Hansen, Hurwitz and Madow (1953), p. 240 (PDF p. 12)", "pdf_page": 12, "printed_page": "240", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "``Hansen, M.H., Hurwitz, W.N. and Madow, W.G. (1953). \\emph{Sample Survey Methods and Theory: Volume 1 \\hl{Methods and Theory}}. New York: John Wiley \\& Sons, Inc.''", "corrected": "``Hansen, M.H., Hurwitz, W.N. and Madow, W.G. (1953). \\emph{Sample Survey Methods and Theory: Volume \\hl{I: Methods and Applications}}. New York: John Wiley \\& Sons, Inc.''", "evidence": "I checked the subtitle against public bibliographic sources. The Helka library catalogue entry for the 1953 New York Wiley edition (\\url{https://www.finna.fi/Record/helka.998190593506253}) reads ``Sample survey methods and theory. Volume 1, Methods and applications'', and the publisher's listing of the 1993 Wiley Classics Library reprint has the same subtitle. I did not consult the book itself.", "problem": "The reference prints the wrong subtitle for Volume I. The book is still identified by its authors, title, volume, year and publisher.", "what_survives": "Everything else is unaffected, since nothing depends on this entry." }, "url": "https://pmaronow.github.io/llm-audit/works/P027/#P027-E009", "work": { "arxiv_id": null, "doi": null, "title": "Conservative Variance Estimation for Sampling Designs with Zero Pairwise Inclusion Probabilities", "url": "https://pmaronow.github.io/llm-audit/works/P027/" }, "work_id": "P027" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Crop of three lines of text and the displayed formula for rho ICC. Red boxes mark the words 'the intra-class correlation over strata', which run from the second line into the third, and the term '1/2' in the denominator 'alpha squared plus 1/2' of the displayed fraction.", "caption": "Supplement, top of p. A2 (supplement PDF p. 2). The marked words call the displayed ratio the intra-class correlation; with the marked $1/2$ it is a variance ratio, and the correlation has 1 in that place.", "url": "https://pmaronow.github.io/llm-audit/works/P027/exhibits/P027-A001.png" } ], "formal_release": null, "formal_statements": [ { "id": "P027-T16", "url": "https://pmaronow.github.io/llm-audit/works/P027/#P027-T16" }, { "id": "P027-T17", "url": "https://pmaronow.github.io/llm-audit/works/P027/#P027-T17" } ], "headline": "Under independent outcome errors, the ordinary same-stratum pair correlation is $\\alpha^2/(\\alpha^2+1)$; the printed $\\alpha^2/(\\alpha^2+1/2)$ is a variance ratio built from the divisor-1 variance $2\\alpha^2$ of the two stratum means.", "id": "P027-A001", "location": "Supplement, Section B, first lines and first display of p. A2 (supplement PDF p. 2); the $\\rho_{ICC}$ labels of Table A1, p. A5, and Figure A1, p. A6", "pdf_page": 2, "printed_page": "A2", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P027-T16", "P027-T17" ], "text": { "as_printed": "``where $\\alpha_s$ is a stratum-specific constant and $\\epsilon_{ks} \\sim \\mathrm{N}(0, 1)$. For the two strata, 1 and 2, we keep things simple by setting $\\alpha_1 = \\alpha = -\\alpha_2$, in which case \\hl{the intra-class correlation over strata} is given by,'' followed by the display\n\\[\n\\rho_{ICC} = \\frac{\\alpha^2}{\\alpha^2 + \\hl{1/2}}.\n\\]", "corrected": "``For the two strata, 1 and 2, we keep things simple by setting $\\alpha_1 = \\alpha = -\\alpha_2$, and the simulations are indexed by \\hl{the variance ratio}\n\\[\n\\rho_{ICC} = \\hl{\\frac{S_\\alpha^2}{S_\\alpha^2+1}} = \\frac{\\alpha^2}{\\alpha^2 + 1/2},\\qquad S_\\alpha^2=\\sum_{s=1}^{2}\\frac{(\\alpha_s-\\bar\\alpha)^2}{2-1}=2\\alpha^2,\n\\]\nwhere $\\bar\\alpha=0$ is the mean of the two stratum constants. Table A1 and Figure A1 use this parameter. Under independent outcome errors, the correlation of two distinct units sharing \\hl{a uniformly chosen stratum} is instead \\hl{$\\alpha^2/(\\alpha^2+1)$}, which equals $\\rho_{ICC}/(2-\\rho_{ICC})$.''", "evidence": "The evidence is a derivation with an exact instance. Let $S=\\pm 1$ with probability $1/2$ each mark the common stratum, and let $Y=\\alpha S+\\epsilon$ and $Y'=\\alpha S+\\epsilon'$, where $\\epsilon$ and $\\epsilon'$ are independent $\\mathrm{N}(0,1)$ variables that are independent of $S$. Then I have $\\mathbb{E}Y=\\mathbb{E}Y'=0$, $\\operatorname{Var}(Y)=\\operatorname{Var}(Y')=\\alpha^2+1$ and $\\operatorname{Cov}(Y,Y')=\\alpha^2$, which implies that\n\\[\n\\operatorname{Corr}(Y,Y')=\\frac{\\alpha^2}{\\alpha^2+1}.\n\\]\nAt $\\alpha=1/2$ this is $1/5$, against the printed $1/3$. The printed ratio equals $S_\\alpha^2/(S_\\alpha^2+1)$, where $S_\\alpha^2=\\sum_{s=1}^{2}(\\alpha_s-\\bar\\alpha)^2/(2-1)=2\\alpha^2$ is the variance of the two stratum means with divisor 1. Solving the printed formula for $\\alpha^2$ and substituting, I obtain $\\operatorname{Corr}(Y,Y')=\\rho_{ICC}/(2-\\rho_{ICC})$, an increasing function of $\\rho_{ICC}$. For the labels 0.33, 0.50, 0.67, 0.82, 0.90 and 0.99 of Table A1, taken as exact, the correlations are 0.198, 0.333, 0.504, 0.695, 0.818 and 0.980.", "problem": "The supplement does not say which convention it uses, and under the ordinary one the value is different. For two distinct units of the same stratum, where the stratum is either of the two with probability $1/2$ and the errors are independent with unit variance, the correlation of the outcomes is $\\alpha^2/(\\alpha^2+1)$. The printed ratio exceeds this by $\\alpha^2/\\bigl((\\alpha^2+1)(2\\alpha^2+1)\\bigr)$, which vanishes only at $\\alpha=0$. The $\\rho_{ICC}$ values that label Table A1 and Figure A1 are not ordinary correlations: the label 0.50 corresponds to a correlation of $1/3$. The printed ratio is a coherent parameter of another kind, and the printed formula still identifies every scenario.", "what_survives": "No simulated value changes. Since the relabelling is monotone, the order of the scenarios, every estimator value in Table A1 and Figure A1 and every comparison within a scenario stand, as does the remark that all approximations perform badly when the parameter is close to 1. A threshold stated in terms of $\\rho_{ICC}$ must be converted if the ordinary correlation is meant. The main text and the simulations of Section C do not use this parameter." }, "url": "https://pmaronow.github.io/llm-audit/works/P027/#P027-A001", "work": { "arxiv_id": null, "doi": null, "title": "Conservative Variance Estimation for Sampling Designs with Zero Pairwise Inclusion Probabilities", "url": "https://pmaronow.github.io/llm-audit/works/P027/" }, "work_id": "P027" }, { "before_review": null, "class": "clarity_note", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "complete_derivation", "exhibits": [ { "alt": "Crop of five lines of the supplement. A red box marks the words 'the empirical variance of mu-hat HT' in the sentence that begins 'For each rho ICC and M value, we used'.", "caption": "Supplement, Section B, p. A2 (supplement PDF p. 2). The marked words describe the “True” column without saying over which draws the variance is taken.", "url": "https://pmaronow.github.io/llm-audit/works/P027/exhibits/P027-A002.png" }, { "alt": "Crop of six lines of the introduction of the supplement. Red boxes mark the words 'computed exactly using expressions from Propositions 1 and 2 in the main text.', which run from the third line into the fourth.", "caption": "Supplement, Section A, p. A1 (supplement PDF p. 1). The marked words say that the expected values of the paper’s two estimators were computed exactly.", "url": "https://pmaronow.github.io/llm-audit/works/P027/exhibits/P027-A002-b.png" } ], "formal_release": null, "formal_statements": [], "headline": "“True” in Tables A1 to A3 targets the sampling variance conditional on each generated population, averaged over the populations; “the empirical variance of $\\hat\\mu_{HT}$” does not say this, and the number of populations behind Tables A2 and A3 is not given.", "id": "P027-A002", "location": "Supplement, Section B, p. A2 (supplement PDF p. 2), paragraph beginning “Table A1 and Figure A1 display”; compare Section A, p. A1, and Section C, pp. A3–A4", "pdf_page": 2, "printed_page": "A2", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": null, "subject": "paper", "target_ids": [], "text": { "as_printed": "Supplement, p.~A2: ``We simulated 1,000 sets of outcomes, $(y_1, y_2)$ over varying values of $M$ and $\\rho_{ICC}$. To each set, we applied the single-unit-per-stratum design.'' Later in the same section: ``For each $\\rho_{ICC}$ and $M$ value, we used \\hl{the empirical variance of $\\hat\\mu_{HT}$} as an estimate of the true variance (Var or ``True''), and means of the computed $(1/N^2)\\widehat{\\operatorname{Var}}(\\hat t)$ \\ldots values as estimates of the expected values of these estimators.''\n\nSupplement, p.~A1: ``In the simulations, expected values of $\\widehat{\\operatorname{Var}}(\\hat t)$ and $\\widehat{\\operatorname{Var}}_C(\\hat t)$ were computed exactly using expressions from Propositions 1 and 2 in the main text.''", "corrected": "``For each $\\rho_{ICC}$ and $M$ value, ``True'' \\hl{targets} the sampling variance of $\\hat\\mu_{HT}$ \\hl{conditional on a generated finite population, averaged over the generated populations}; it is not the pooled variance obtained by drawing a fresh population and one sample from it at every repetition.'' The supplement should also state how each conditional variance was calculated or estimated and how many populations were generated for Tables A2 and A3. I could not establish these details without the simulation code, which was unavailable to me.", "intended_meaning": "The simulations concern the sampling variance for fixed outcomes. I therefore read ``True'' as targeting the variance of $\\hat\\mu_{HT}$ over samples for one generated population, averaged over the generated populations. Table A1 agrees with this reading and not with a variance pooled over fresh populations. With two strata of size $M$ and independent unit-variance errors the expected conditional variance is $(M-1)/(2M)$, which is 0.25 at $M=2$ as printed. The pooled variance is $1/2$. On this reading the exact computation within each population described on p.~A1 and the averages over populations described on p.~A2 are compatible. The supplement does not say how the conditional variance was obtained for each population. Pages A3--A4 do not state the number of populations behind Tables A2 and A3; the ``1000 simulated sample draws'' mentioned there concern the with-replacement estimator. No table entry is shown to be affected." }, "url": "https://pmaronow.github.io/llm-audit/works/P027/#P027-A002", "work": { "arxiv_id": null, "doi": null, "title": "Conservative Variance Estimation for Sampling Designs with Zero Pairwise Inclusion Probabilities", "url": "https://pmaronow.github.io/llm-audit/works/P027/" }, "work_id": "P027" }, { "before_review": { "class": "error", "severity": "Moderate" }, "class": "clarity_note", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Crop of the four-line paragraph that begins 'If any units k, l are in clusters'. Red boxes mark the words 'Combining units will tend to reduce the bias of the variance estimator', which run from the second line into the third.", "caption": "Section 3.1, last paragraph, p. 237 (PDF p. 9). The marked words state that combining units will tend to reduce the bias of the variance estimator.", "url": "https://pmaronow.github.io/llm-audit/works/P027/exhibits/P027-N002.png" } ], "formal_release": null, "formal_statements": [ { "id": "P027-T3", "url": "https://pmaronow.github.io/llm-audit/works/P027/#P027-T3" }, { "id": "P027-T4", "url": "https://pmaronow.github.io/llm-audit/works/P027/#P027-T4" } ], "headline": "Totalling units that are always selected together need not reduce the bias of the corrected variance estimator: with $a_{kl}=b_{kl}=2$ it can raise the bias when the clusters differ in size, and it cannot raise the bias when every two clusters with zero joint inclusion probability have the same size.", "id": "P027-N002", "location": "Section 3.1, paragraph after the proof of Corollary 1, p. 237 (PDF p. 9)", "pdf_page": 9, "printed_page": "237", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": null, "subject": "paper", "target_ids": [ "P027-T3", "P027-T4" ], "text": { "as_printed": "``If any units $k, l$ are in clusters (\\emph{i.e.}, $\\Pr(I_k \\neq I_l) = 0$), these units should be totaled into one larger unit before estimation. Combining units \\hl{will tend to reduce the bias} of the variance estimator because only pairs of cluster-level totals will be included in $A^*$, as opposed to all constituent pairs.''", "corrected": "An optional clarification after the original recommendation is:\n\n``For $a_{kl}=b_{kl}=2$, the prediction that totalling will tend to reduce bias is \\hl{a heuristic, not a guarantee}. Totalling can increase the bias when cluster sizes differ. If every two clusters with zero joint inclusion probability have \\hl{equal sizes}, the bias does not increase.''", "intended_meaning": "The paragraph gives a heuristic reason to expect smaller bias. The words ``will tend to'' allow exceptions; they do not assert that totalling reduces bias for every admissible design and fixed outcome vector. The number of pairs does not settle the direction of the change, because totalling also replaces sums of squared outcomes by squared totals, and these can be larger. For the choice $a_{kl}=b_{kl}=2$ of Section 3.2, totalling leaves the Horvitz--Thompson total, its variance, the uncorrected variance estimator and the term $A$ unchanged. The whole change in the bias of $\\widehat{\\operatorname{Var}}_{C2}(\\hat t)$ then comes from $A^*$, and this change can have either sign. In a population of three units with outcomes $(1,1,1)$, in which units 1 and 2 are always selected together, the bias rises from 8 to 9 after totalling. When all outcomes equal the same nonzero constant, totalling raises the bias whenever two clusters with zero joint inclusion probability differ in size. Designs that draw one cluster per stratum are of this kind when clusters within a stratum differ in size. The paragraph specifies neither a class of populations nor a distribution defining this tendency. The counterexample establishes a limitation of the advice, not falsity of its hedged prediction.\n\nThe evidence is a derivation and an exact counterexample. I partition the population into groups $G_g$ of units that are always selected together, and I write $n_g$ for their sizes, $Y_g=\\sum_{k\\in G_g}y_k$ for their totals and $S_g=\\sum_{k\\in G_g}y_k^2$ for their sums of squares. Let $\\mathcal{E}$ be the set of unordered pairs $\\{g,h\\}$ of groups with zero joint inclusion probability. For $a_{kl}=b_{kl}=2$, summing the terms of $A^*$ over each pair of groups gives\n\\[\nA^*_{\\text{units}}=\\sum_{\\{g,h\\}\\in\\mathcal{E}}\\left(n_hS_g+n_gS_h\\right),\\qquad A^*_{\\text{totalled}}=\\sum_{\\{g,h\\}\\in\\mathcal{E}}\\left(Y_g^2+Y_h^2\\right).\n\\]\nSince $A$ is the same in both cases, the bias $A+A^*$ before totalling minus the bias after totalling is\n\\[\n\\sum_{\\{g,h\\}\\in\\mathcal{E}}\\left(n_hS_g+n_gS_h-Y_g^2-Y_h^2\\right).\n\\]\nIf $y_k=v$ for all $k$, then $S_g=n_gv^2$ and $Y_g=n_gv$, and this difference equals $-v^2\\sum_{\\{g,h\\}\\in\\mathcal{E}}(n_g-n_h)^2\\le 0$. If $n_g=n_h$ for every pair in $\\mathcal{E}$, each summand is nonnegative, because $n_gS_g\\ge Y_g^2$.\n\nIn the three-unit example the possible samples are $\\{1,2\\}$ and $\\{3\\}$, each with probability $1/2$, and $y=(1,1,1)$. The true variance of $\\hat t$ is 1.\n\n\\begin{audittable}{llrrrrr}{The three-unit example, enumerated in full. The bias of the corrected estimator is $9-1=8$ on the original units and $10-1=9$ after units 1 and 2 are totalled.}\nSample & Probability & $\\hat t$ & $\\widehat A^*$, units & $\\widehat A^*$, totalled & $\\widehat{\\operatorname{Var}}_{C2}$, units & $\\widehat{\\operatorname{Var}}_{C2}$, totalled \\\\\n$\\{1,2\\}$ & $1/2$ & 4 & 4 & 8 & 12 & 16 \\\\\n$\\{3\\}$ & $1/2$ & 2 & 4 & 2 & 6 & 4 \\\\\nExpectation & & 3 & 4 & 5 & 9 & 10 \\\\\n\\end{audittable}\n\nFurther exact cases show both directions. In each, exactly one cluster is selected, with probability $1/2$ unless stated, and a bar separates the clusters.\n\n\\begin{audittable}{lrrr}{Bias of $\\widehat{\\operatorname{Var}}_{C2}(\\hat t)$ before and after totalling, by exact enumeration.}\nOutcomes by cluster & $\\operatorname{Var}(\\hat t)$ & Bias, units & Bias, totalled \\\\\n$(1,1\\mid 1)$ & 1 & 8 & 9 \\\\\n$(1,1\\mid 0)$ & 4 & 2 & 4 \\\\\n$(1,1\\mid -1/4)$ & $81/16$ & $9/8$ & $49/16$ \\\\\n$(1,1,1,1\\mid 1)$ & 9 & 16 & 25 \\\\\n$(3,1,2,5,4\\mid 2)$, probabilities $1/3$ and $2/3$ & 392 & 135 & 289 \\\\\n$(1,-1\\mid 2,-2)$ & 0 & 20 & 0 \\\\\n\\end{audittable}\n\nPropositions 1 and 2 and Corollaries 1 to 3 are unaffected. The estimator $\\widehat{\\operatorname{Var}}_{C2}(\\hat t)$ is conservative with and without totalling, and no proof uses this paragraph. The practical advice is heuristic; the example supplies an important exception without refuting the claimed tendency. The phrase ``will tend to'' is not refuted as a statement about typical populations, since the paper specifies no distribution of outcomes or cluster sizes, and totalling does reduce the bias in some cases. Since the groups in the periodic designs of the supplement have equal sizes, totalling cannot raise the bias there (see \\fid{P027-N005})." }, "url": "https://pmaronow.github.io/llm-audit/works/P027/#P027-N002", "work": { "arxiv_id": null, "doi": null, "title": "Conservative Variance Estimation for Sampling Designs with Zero Pairwise Inclusion Probabilities", "url": "https://pmaronow.github.io/llm-audit/works/P027/" }, "work_id": "P027" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "complete_derivation", "exhibits": [ { "alt": "Crop of rows 29 to 60 of a table of numbers with fifteen columns. Red boxes mark, in rows 30, 40, 50 and 60, the two adjacent cells that hold the mean of the uncorrected estimator and the mean of the corrected estimator: 1412.15 and 2029.28, 1564.34 and 2396.58, 1430.59 and 2032.07, 1562.38 and 2395.86.", "caption": "Supplement Table A3, p. A8 (supplement PDF p. 8), rows 29 to 60. The columns are the row number, $N$, $m$, $\\rho$, $\\bar y$, the true variance, the means of $\\widehat{\\operatorname{Var}}_{HT}$, $\\widehat{\\operatorname{Var}}_{C2}$ and $\\widehat{\\operatorname{Var}}_{WR}$, their relative biases and their nominal biases. The marked cells are the means of $\\widehat{\\operatorname{Var}}_{HT}$ and $\\widehat{\\operatorname{Var}}_{C2}$ in rows 30, 40, 50 and 60.", "url": "https://pmaronow.github.io/llm-audit/works/P027/exhibits/P027-N004.png" } ], "formal_release": null, "formal_statements": [ { "id": "P027-T9", "url": "https://pmaronow.github.io/llm-audit/works/P027/#P027-T9" }, { "id": "P027-T19", "url": "https://pmaronow.github.io/llm-audit/works/P027/#P027-T19" } ], "headline": "In the four rows of Table A3 with $\\rho=0.99$ and $N\\ge 120$, the printed means of $\\widehat{\\operatorname{Var}}_{HT}$ and $\\widehat{\\operatorname{Var}}_{C2}$ are overwhelmingly inconsistent with the exact-expectation simulation that Section A describes; the expectations under the stated model suggest that the $\\widehat{\\operatorname{Var}}_{HT}$ means are 1 to 5 per cent too high.", "id": "P027-N004", "location": "Supplement Table A3, p. A8 (supplement PDF p. 8), rows 30, 40, 50 and 60, columns $\\widehat{\\operatorname{Var}}_{HT}$ and $\\widehat{\\operatorname{Var}}_{C2}$ under “Mean from sims.”; compare Section A, p. A1", "pdf_page": 8, "printed_page": "A8", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P027-T19", "P027-T9" ], "text": { "as_printed": "Supplement, p.~A1: ``In the simulations, expected values of $\\widehat{\\operatorname{Var}}(\\hat t)$ and $\\widehat{\\operatorname{Var}}_C(\\hat t)$ were computed exactly using expressions from Propositions 1 and 2 in the main text.''\n\nSupplement, Table A3 (``Simulation results for PPS systematic sampling: periodic size case''), four of its rows:\n\n\\begin{audittable}{rrrrrrrrr}{Rows 30, 40, 50 and 60 of Table A3 as printed: the true variance, and the mean and the relative bias of $\\widehat{\\operatorname{Var}}_{HT}$ and of $\\widehat{\\operatorname{Var}}_{C2}$.}\nRow & $N$ & $m$ & $\\rho$ & True Var & Mean, HT & Mean, C2 & Rel. bias, HT & Rel. bias, C2 \\\\\n30 & 120 & 30 & 0.99 & 1578.87 & \\hl{1412.15} & \\hl{2029.28} & $-0.11$ & 0.29 \\\\\n40 & 600 & 30 & 0.99 & 1587.01 & \\hl{1564.34} & \\hl{2396.58} & $-0.01$ & 0.51 \\\\\n50 & 240 & 60 & 0.99 & 1581.13 & \\hl{1430.59} & \\hl{2032.07} & $-0.10$ & 0.29 \\\\\n60 & 1200 & 60 & 0.99 & 1586.32 & \\hl{1562.38} & \\hl{2395.86} & $-0.02$ & 0.51 \\\\\n\\end{audittable}", "corrected": "A note under Table A3: ``In rows 30, 40, 50 and 60 the means of $\\widehat{\\operatorname{Var}}_{HT}$ and $\\widehat{\\operatorname{Var}}_{C2}$ are \\hl{statistically incompatible with averages of exact design expectations} for the design and outcome model of Section C with \\hl{independent Gaussian errors}; the lower-tail probability for each row is below $10^{-56}$. Under that design and model the ratio of the expected bias of $\\widehat{\\operatorname{Var}}_{HT}$ to the expected true variance is \\hl{$-5/36$} when $N/m=4$ and \\hl{$-1/36$} when $N/m=20$.''\n\nFor the stated design, these entries and the matching points of Figure A3 are to be regenerated from the original populations with the exact probabilities $p_r=1/(3q)$ and $5/(3q)$ of the $q$ possible samples, and the number of generated populations is to be stated. If the simulation was run differently, with another ordering of the sizes, another noise model, or averages of simulated estimates in place of exact expectations, then the description in Sections A and C needs the correction instead.", "evidence": "The evidence is a derivation with explicit probability bounds; no simulation was run. Let $q=N/m$, and let $z$ be the demeaned outcomes, $Y_r$ their totals over the $q$ possible samples, $p_r$ the probabilities of these samples and $Q=\\sum_iz_i^2$. On the scale of the mean, the true variance, the expectation $H$ of the uncorrected estimator and the expectation $C$ of the unit-level corrected estimator are\n\\[\nV=\\frac{1}{N^2}\\sum_r\\frac{Y_r^2}{p_r},\\qquad H=V-\\frac{1}{N^2}\\sum_rY_r^2,\\qquad C=H+\\frac{(N-m)Q}{N^2}.\n\\]\nI set $R=5(q-1)/(9q-5)$ and $T=C-(1+R)H$. Decomposing the demeaned vectors into four orthogonal subspaces diagonalizes $T$, and its coefficient on the direction $x-6$ is zero. With $z=\\beta(x-6)+\\epsilon-\\bar\\epsilon$ and independent $\\mathrm{N}(0,\\sigma^2)$ errors, $\\sigma^2=900(1-\\rho^2)$, the signal and its cross terms with the noise drop out, and I obtain\n\\[\nT=a\\left[X_W+c\\left(X_G-X_L\\right)\\right],\\qquad a=\\frac{(N-m)\\sigma^2}{N^2},\\quad c=\\frac{6q}{9q-5},\n\\]\nwhere $X_W$, $X_G$ and $X_L$ are independent chi-squared variables with $N-q$, $q/2-1$ and $q/2-1$ degrees of freedom. This implies that $\\mathbb{E}T=a(N-q)>0$ and $T\\ge -acX_L$. A Markov bound on $\\exp(X_L/4)$ then bounds the probability of the printed values, allowing an error of 0.005 in each printed mean and any number $K\\ge 1$ of generated populations.\n\n\\begin{audittable}{rrrrrrr}{The four rows: the contrast $T$ formed from the printed means, the upper bound on its probability, and the expectations $\\mathbb{E}H$ and $\\mathbb{E}C$ under the stated model beside the printed means of $\\widehat{\\operatorname{Var}}_{HT}$ and $\\widehat{\\operatorname{Var}}_{C2}$.}\nRow & $T$ & Probability bound & $\\mathbb{E}H$, model & Mean, HT & $\\mathbb{E}C$, model & Mean, C2 \\\\\n30 & $-66.17$ & $10^{-56}$ & 1356.54 & 1412.15 & 2025.91 & 2029.28 \\\\\n40 & $-16.97$ & $10^{-64}$ & 1542.08 & 1564.34 & 2395.65 & 2396.58 \\\\\n50 & $-90.74$ & $10^{-157}$ & 1361.89 & 1430.59 & 2034.08 & 2032.07 \\\\\n60 & $-14.67$ & $10^{-111}$ & 1542.87 & 1562.38 & 2397.15 & 2395.86 \\\\\n\\end{audittable}\n\nThe model expectations use the sample standard deviation of $x$ in $\\beta$; they are not replacements for the unknown realized averages. The bound does not use the ``True'' or with-replacement columns. It is a statement of probability and not of impossibility, since Gaussian noise is unbounded. For the other rows examined the same bound is uninformative (rows 20, 21 and 41) or weak (row 10, at most $2^{-4}$).", "problem": "Section A says that the expected values of the two estimators were computed exactly for each generated population. In the periodic design one combination of the two expectations does not depend on the signal $\\beta x$ at all: it is a known quadratic form in the Gaussian noise, with positive mean. In rows 30, 40, 50 and 60 the combination formed from the printed means equals $-66.17$, $-16.97$, $-90.74$ and $-14.67$. With independent Gaussian errors at the stated scale, values at least this negative have probability below $10^{-56}$, whatever the number of generated populations and after allowing for rounding. I conclude that at least one of the two columns, or the description of how they were computed, is wrong in these rows. It is not settled which. The expectations under the stated model are close to the printed $\\widehat{\\operatorname{Var}}_{C2}$ means and 1 to 5 per cent below the printed $\\widehat{\\operatorname{Var}}_{HT}$ means. On that account the table understates the negative bias of the uncorrected estimator in these rows: it prints relative biases of $-0.11$, $-0.01$, $-0.10$ and $-0.02$, against about $-0.14$ when $N/m=4$ and $-0.03$ when $N/m=20$.", "what_survives": "Proposition 1 and the conservativeness of the corrected estimator stand; the check uses them. The qualitative comparison in the affected rows of Table A3 survives: the uncorrected estimator is slightly below the true variance, the corrected estimator is above it, and the with-replacement estimator is far below it. Tables A1 and A2 are not touched by this check. No corrected entries are supplied, because the generated populations and the code are not available. The explanation that these entries appear to support is qualified in \\fid{P027-E004}." }, "url": "https://pmaronow.github.io/llm-audit/works/P027/#P027-N004", "work": { "arxiv_id": null, "doi": null, "title": "Conservative Variance Estimation for Sampling Designs with Zero Pairwise Inclusion Probabilities", "url": "https://pmaronow.github.io/llm-audit/works/P027/" }, "work_id": "P027" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Crop of four lines of the main text. Red boxes mark the words 'these units should be totaled into one larger unit before estimation.', which run from the end of the second line into the third.", "caption": "Main text, Section 3.1, p. 237 (PDF p. 9). The marked words are the instruction to total units that are always selected together.", "url": "https://pmaronow.github.io/llm-audit/works/P027/exhibits/P027-N005.png" }, { "alt": "Crop of four lines of the supplement. Red boxes mark the words 'is generally quite conservative.', which follow the symbol for the corrected variance estimator.", "caption": "Supplement, Section C, p. A4 (supplement PDF p. 4). The marked words describe the $\\widehat{\\operatorname{Var}}_{C2}$ results of the periodic simulation.", "url": "https://pmaronow.github.io/llm-audit/works/P027/exhibits/P027-N005-b.png" }, { "alt": "Crop of rows 20 to 52 of a table of numbers with fifteen columns. Red boxes mark, in rows 21, 31, 41 and 51, the mean of the corrected estimator (653.92, 891.43, 622.28, 861.11) and its relative bias (14.37, 14.82, 25.89, 29.13).", "caption": "Supplement Table A3, p. A8 (supplement PDF p. 8), rows 20 to 52. The columns are the row number, $N$, $m$, $\\rho$, $\\bar y$, the true variance, the means of $\\widehat{\\operatorname{Var}}_{HT}$, $\\widehat{\\operatorname{Var}}_{C2}$ and $\\widehat{\\operatorname{Var}}_{WR}$, their relative biases and their nominal biases. The marked cells are the mean and the relative bias of $\\widehat{\\operatorname{Var}}_{C2}$ in rows 21, 31, 41 and 51.", "url": "https://pmaronow.github.io/llm-audit/works/P027/exhibits/P027-N005-c.png" }, { "alt": "Two panels, one for sampling fraction one quarter and one for one twentieth, plot relative bias on a logarithmic axis with labelled ticks from 0.1 to 30 against rho from 0 to 1. Blue open markers joined by lines are the printed values for three designs in each panel. For the designs with m equal to 30 or 60 they fall from about 14 to 29 at rho 0.05 to about 0.3 or 0.5 at rho 0.99. A grey band marks the range possible after totalling, one sixth to five sixths in the left panel and three tenths to three halves in the right panel, and an orange dashed line marks the ratio of outcome-averaged bias to outcome-averaged variance, five eighteenths on the left and one half on the right. The printed values for the designs with m equal to 2 lie inside the band.", "caption": "Relative bias of $\\widehat{\\operatorname{Var}}_{C2}$ in the 60 rows of supplement Table A3, formed as (printed mean $-$ printed true variance)/printed true variance, on a logarithmic scale. The shaded band is the range that the estimator can reach after always-co-selected units are totalled; the dashed line is the ratio of outcome-averaged bias to outcome-averaged variance derived for the totalled estimator under the stated outcome model. The printed values for the designs with $m=30$ and $m=60$ lie far above the band at low $\\rho$.", "url": "https://pmaronow.github.io/llm-audit/works/P027/exhibits/P027-N005-c2-relative-bias.png" } ], "formal_release": null, "formal_statements": [ { "id": "P027-T5", "url": "https://pmaronow.github.io/llm-audit/works/P027/#P027-T5" }, { "id": "P027-T6", "url": "https://pmaronow.github.io/llm-audit/works/P027/#P027-T6" }, { "id": "P027-T9", "url": "https://pmaronow.github.io/llm-audit/works/P027/#P027-T9" } ], "headline": "The $\\widehat{\\operatorname{Var}}_{C2}$ results of the periodic simulation (Table A3 and Figure A3) are for the estimator applied to the original units, without the totalling of always-co-selected units that p. 237 of the main text recommends, and the supplement should say so.", "id": "P027-N005", "location": "Main text, Section 3.1, p. 237 (PDF p. 9), paragraph beginning “If any units”; supplement, Section C, p. A4 (supplement PDF p. 4); supplement Table A3, p. A8 (supplement PDF p. 8), columns $\\widehat{\\operatorname{Var}}_{C2}$", "pdf_page": 9, "printed_page": "237", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Moderate", "subject": "paper", "target_ids": [ "P027-T5", "P027-T6", "P027-T9" ], "text": { "as_printed": "Main text, p.~237: ``If any units $k, l$ are in clusters (\\emph{i.e.}, $\\Pr(I_k \\neq I_l) = 0$), these units should be totaled into one larger unit before estimation.''\n\nSupplement, p.~A4: ``As expected, $\\widehat{\\operatorname{Var}}_{WR}(\\hat\\mu_{HT})$ tends to greatly understate the sampling variance of $\\hat\\mu_{HT}$, while $\\widehat{\\operatorname{Var}}_{C2}(\\hat t)$ is generally quite conservative.''\n\nSupplement, Table A3 (``Simulation results for PPS systematic sampling: periodic size case.''), four of its rows:\n\n\\begin{audittable}{rrrrrrrr}{Rows 21, 31, 41 and 51 of Table A3 as printed: the true variance, the means of $\\widehat{\\operatorname{Var}}_{HT}$ and $\\widehat{\\operatorname{Var}}_{C2}$, and the relative bias of $\\widehat{\\operatorname{Var}}_{C2}$.}\nRow & $N$ & $m$ & $\\rho$ & True Var & Mean, HT & Mean, C2 & Rel. bias, C2 \\\\\n21 & 120 & 30 & 0.05 & 42.55 & 36.78 & \\hl{653.92} & \\hl{14.37} \\\\\n31 & 600 & 30 & 0.05 & 56.35 & 54.90 & \\hl{891.43} & \\hl{14.82} \\\\\n41 & 240 & 60 & 0.05 & 23.14 & 20.39 & \\hl{622.28} & \\hl{25.89} \\\\\n51 & 1200 & 60 & 0.05 & 28.58 & 27.82 & \\hl{861.11} & \\hl{29.13} \\\\\n\\end{audittable}", "corrected": "Supplement, p.~A4: ``As expected, $\\widehat{\\operatorname{Var}}_{WR}(\\hat\\mu_{HT})$ tends to greatly understate the sampling variance of $\\hat\\mu_{HT}$, while $\\widehat{\\operatorname{Var}}_{C2}(\\hat t)$ is generally quite conservative. The $\\widehat{\\operatorname{Var}}_{C2}$ results here \\hl{illustrate the estimator applied to the original units, rather than the estimator after combining always-co-selected units} as recommended on p.~237 of the main text.'' The same note belongs under Table A3 and Figure A3.\n\nFor the totalled estimator in these designs, with $q=N/m$ equal to 4 or 20 and demeaned outcomes, the expectation on the scale of the mean is $\\bigl(V+(q-2)S\\bigr)/N^2$, and its relative bias, \\hl{when $V>0$}, lies between $(q-2)/(3q)$ and $5(q-2)/(3q)$.", "evidence": "The evidence is a derivation, a bound that holds for every population, and an exact example. Let $z_k=\\tilde y_k$ be the demeaned outcomes. Since the sizes alternate between 2 and 10 and $q$ is even, the sample is one of the groups $G_c=\\{c,c+q,\\dots,c+(m-1)q\\}$, $c=1,\\dots,q$, which are selected with probability $p_c=1/(3q)$ for odd $c$ and $p_c=5/(3q)$ for even $c$. I define $Y_c=\\sum_{k\\in G_c}z_k$, $S=\\sum_cY_c^2$ and $Q=\\sum_kz_k^2$. On the scale of the total, I have $V=\\operatorname{Var}(\\hat t)=\\sum_cY_c^2/p_c$, the uncorrected estimator has expectation $H=V-S$, and\n\\[\n\\mathbb{E}\\,\\widehat{\\operatorname{Var}}_{C2}(\\hat t)=V-S+m(q-1)Q\\ \\text{ on the original units},\\qquad \\mathbb{E}\\,\\widehat{\\operatorname{Var}}_{C2}(\\hat t)=V+(q-2)S\\ \\text{ after totalling}.\n\\]\nThe first exceeds the second by $(q-1)(mQ-S)\\ge 0$. Since $3qS/5\\le V\\le 3qS$, for $V>0$ the relative bias $(q-2)S/V$ of the totalled estimator satisfies\n\\[\n\\frac{q-2}{3q}\\le\\frac{(q-2)S}{V}\\le\\frac{5(q-2)}{3q}.\n\\]\nAllowing an error of 0.005 in each printed entry, the printed $\\widehat{\\operatorname{Var}}_{C2}$ mean lies above the upper bound in 26 of the 60 rows, all with $m\\ge 30$ and $\\rho\\le 0.6$. Under the stated outcome model with fixed $\\beta$ and independent mean-zero errors of equal variance, the ratio of the outcome-averaged bias of the totalled estimator to the outcome-averaged variance is $(5/9)(1-2/q)$, that is, $5/18$ for $q=4$ and $1/2$ for $q=20$.\n\nAn exact example for $N=120$, $m=30$ takes $z_k=x_k-6+2\\lfloor k/4\\rfloor-29$ for $k=0,\\dots,119$, with $x_k=2$ for even $k$ and $x_k=10$ for odd $k$. On the scale of the mean the true variance is $144/5=28.8$, the unit-level estimator has expectation $5231/20=261.55$, and the totalled estimator has expectation $184/5=36.8$.\n\nSince $S=V-H$, the totalled estimator has expectation $V+(q-2)(V-H)$. If the printed true variance and the printed mean of $\\widehat{\\operatorname{Var}}_{HT}$ are averages of $V$ and $H$ over the same populations, then the mean of the totalled estimator follows from those two columns.\n\n\\begin{audittable}{rrrrrr}{The four rows quoted above: the printed relative bias of $\\widehat{\\operatorname{Var}}_{C2}$, the range possible for the totalled estimator, and the mean and relative bias of the totalled estimator implied by the printed true variance and the printed mean of $\\widehat{\\operatorname{Var}}_{HT}$.}\nRow & $q$ & Printed rel. bias & Range after totalling & Implied mean & Implied rel. bias \\\\\n21 & 4 & 14.37 & $1/6$ to $5/6$ & 54.09 & 0.27 \\\\\n31 & 20 & 14.82 & $3/10$ to $3/2$ & 82.45 & 0.46 \\\\\n41 & 4 & 25.89 & $1/6$ to $5/6$ & 28.64 & 0.24 \\\\\n51 & 20 & 29.13 & $3/10$ to $3/2$ & 42.26 & 0.48 \\\\\n\\end{audittable}\n\nOn these implied values the totalled estimator is closer to the true variance than the original-unit with-replacement estimator in 41 of the 60 rows; the printed column is closer in 20. I do not have the simulation code or the generated populations, so I cannot confirm that the printed entries are correctly computed values of the unit-level estimator.", "problem": "In each of the six designs of Table A3 the systematic sample is one of $q=N/m$ disjoint groups of $m$ units, and all units of a group are always selected together. This is the case in which p.~237 says that the units ``should be totaled into one larger unit before estimation.'' The $\\widehat{\\operatorname{Var}}_{C2}$ entries of Table A3 cannot be values of the estimator after such totalling. For any demeaned population with positive design variance the relative bias of the totalled estimator lies between $1/6$ and $5/6$ when $q=4$ and between $3/10$ and $3/2$ when $q=20$. The table prints 14.37, 14.82, 25.89 and 29.13 in the rows shown above. I therefore read these entries as results for the estimator applied to the $N$ original units; the supplement does not disclose this departure from the totalling recommendation. Taken as the performance of the procedure that the main text recommends, Table A3 and Figure A3 overstate its bias at low $\\rho$ by one to two orders of magnitude. The unit-level estimator is conservative, and ``generally quite conservative'' correctly describes the displayed values.", "what_survives": "The Horvitz--Thompson estimator, its variance and the uncorrected variance estimator are the same with and without totalling, and $\\widehat{\\operatorname{Var}}_{C2}$ is conservative in both forms. I retain the original-unit with-replacement estimator as the comparator. The claim on p.~240 of the main text, that the estimators perform favourably under substantial periodicity, is not weakened: in these designs totalling cannot raise the bias, and at $\\rho=0.99$ the printed relative biases (0.28 to 0.51) are close to the outcome-averaged ratios $5/18$ and $1/2$ derived for the totalled estimator. Only the reading of Table A3, Figure A3 and the sentence on p.~A4 changes: they describe the unit-level estimator, not the procedure recommended on p.~237. That recommendation is itself qualified in \\fid{P027-N002}." }, "url": "https://pmaronow.github.io/llm-audit/works/P027/#P027-N005", "work": { "arxiv_id": null, "doi": null, "title": "Conservative Variance Estimation for Sampling Designs with Zero Pairwise Inclusion Probabilities", "url": "https://pmaronow.github.io/llm-audit/works/P027/" }, "work_id": "P027" } ], "lean": { "applicable": true, "guide": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/docs/works/P027.md", "repository": "https://github.com/pmaronow/llm-audit-lean", "targets": [ { "declarations": [ { "claim_revision_id": "catalog:8b2c95c8ef410a853bfd085a3fe48ddef3cf843d4fc45853d177159135bde5a8", "conditional_on": [], "credit_reason": "Fidelity sweep: ok", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P027.fourPhase_pi", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P027/P027_Audit.lean#L483", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:8b2c95c8ef410a853bfd085a3fe48ddef3cf843d4fc45853d177159135bde5a8", "conditional_on": [], "credit_reason": "Fidelity sweep: ok", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P027.fourPhase_joint", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P027/P027_Audit.lean#L489", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:8b2c95c8ef410a853bfd085a3fe48ddef3cf843d4fc45853d177159135bde5a8", "conditional_on": [], "credit_reason": "Fidelity sweep: ok", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P027.periodic8_positive", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P027/P027_Audit.lean#L496", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:8b2c95c8ef410a853bfd085a3fe48ddef3cf843d4fc45853d177159135bde5a8", "conditional_on": [], "credit_reason": "Fidelity sweep: ok", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P027.periodic8_bias", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P027/P027_Audit.lean#L504", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:8b2c95c8ef410a853bfd085a3fe48ddef3cf843d4fc45853d177159135bde5a8", "conditional_on": [], "credit_reason": "Fidelity sweep: ok", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P027.periodic8_sample_values", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P027/P027_Audit.lean#L510", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:8b2c95c8ef410a853bfd085a3fe48ddef3cf843d4fc45853d177159135bde5a8", "conditional_on": [], "credit_reason": "Fidelity sweep: ok", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P027.periodic8_counterexample", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P027/P027_Audit.lean#L524", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:8b2c95c8ef410a853bfd085a3fe48ddef3cf843d4fc45853d177159135bde5a8", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': 'P027/P027_Audit.lean:740 identifies the selected subset for every start in each of the four half-open intervals in T1(C1). Those intervals are [0,2), [2,12), [12,14), and [14,24), so they cover the entire claimed start domain, including its internal boundaries. The periodic8 weights at line 462 are exactly their normalized lengths, 1/12, 5/12, 1/12, and 5/12. Representing this uniform-start law by its four finite atoms loses no start or sampling-probability case.'}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P027.periodic8_systematic", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P027/P027_Audit.lean#L740", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:8b2c95c8ef410a853bfd085a3fe48ddef3cf843d4fc45853d177159135bde5a8", "conditional_on": [], "credit_reason": "Fidelity sweep: ok", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P027.four_phase_zero_pair_count", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P027/P027_Audit.lean#L1351", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:8b2c95c8ef410a853bfd085a3fe48ddef3cf843d4fc45853d177159135bde5a8", "conditional_on": [], "credit_reason": "This is an explicit fixed-outcome witness, not a statement about the probability of an exact Gaussian draw. For general q the phase law is the same systematic law: each block has size sum 6q and the start intervals have lengths 2 or 10. The record’s sentence “the theorem remains unproved” conflicts with the current packet status proved; proof checking is outside this review.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P027.alternating_family_witness", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P027/P027_Audit.lean#L2361", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:8b2c95c8ef410a853bfd085a3fe48ddef3cf843d4fc45853d177159135bde5a8", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P027-T1", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P027-T1" }, "html": { "body": "

Supplement pp. A3–A4 and Table A3 (p. A8), with A from main p. 234: the full eight-unit periodic witness has A=-16, the stated samplewise values and moments, downward bias -1/4 and relative bias -5/36; C2 remains conservative and WR is zero. The same alternating populations at every listed size have A=-N²/q and mean-scale bias -1/q, which do not vanish at fixed q. The cumulative-size uniform-start law and demeaning are included explicitly. The usual measure-theoretic conditions are assumed.

", "headline": "Supplement Section C, last paragraph: ‘This occurs because of the special nature of the data generating process, which results in the elements in the bias term, AAA, canceling each other out, in which case AAA tends to 0’, for the periodic design of Table A3; the witness is the validator's construction and is not in the paper", "location": "Supplement p. A4 (staged/supplements/smj_suppl.pdf, PDF p. 4), Section C, last paragraph before the References; the periodic design is described on supplement pp. A3–A4 (PDF pp. 3–4); Table A3 is on supplement p. A8 (PDF p. 8); AAA is defined on main p. 234 (PDF p. 6)" }, "id": "P027-T1", "text": { "body": "Supplement pp. A3–A4 and Table A3 (p. A8), with A from main p. 234: the full eight-unit periodic witness has A=-16, the stated samplewise values and moments, downward bias -1/4 and relative bias -5/36; C2 remains conservative and WR is zero. The same alternating populations at every listed size have A=-N²/q and mean-scale bias -1/q, which do not vanish at fixed q. The cumulative-size uniform-start law and demeaning are included explicitly. The usual measure-theoretic conditions are assumed.", "headline": "Supplement Section C, last paragraph: ‘This occurs because of the special nature of the data generating process, which results in the elements in the bias term, $A$, canceling each other out, in which case $A$ tends to 0’, for the periodic design of Table A3; the witness is the validator's construction and is not in the paper", "location": "Supplement p. A4 (staged/supplements/smj_suppl.pdf, PDF p. 4), Section C, last paragraph before the References; the periodic design is described on supplement pp. A3–A4 (PDF pp. 3–4); Table A3 is on supplement p. 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Supplement pp. A3–A4, Section C, and main p. 234: the centered periodic cumulative-size PPS design has q disjoint phases with the stated probabilities, A=-sum(T_g²), and exactly the stated nonpositive design bias and relative-bias bounds. Under mean-zero uncorrelated finite-variance errors the second moments and both ratios of outcome-averaged quantities are as displayed. The printed calibration follows from the empirical size variance; at fixed q and nonzero rho its bias is uniformly separated from zero and tends to -900 rho²/q. Demeaning does not cancel the bias. Ratios have the positive variance guards in the claim. The usual measure-theoretic conditions are assumed.

", "headline": "Supplement Section C, last paragraph (the explanation of why the uncorrected estimator is close to the true variance in the periodic design), corrected: the bias term is minus the sum of squared sample totals", "location": "Supplement p. A4 (staged/supplements/smj_suppl.pdf, PDF p. 4), Section C, last paragraph before the References; the design on supplement pp. A3–A4; Proposition 1 on main p. 234 (PDF p. 6)" }, "id": "P027-T2", "text": { "body": "Supplement pp. A3–A4, Section C, and main p. 234: the centered periodic cumulative-size PPS design has q disjoint phases with the stated probabilities, A=-sum(T_g²), and exactly the stated nonpositive design bias and relative-bias bounds. Under mean-zero uncorrelated finite-variance errors the second moments and both ratios of outcome-averaged quantities are as displayed. The printed calibration follows from the empirical size variance; at fixed q and nonzero rho its bias is uniformly separated from zero and tends to -900 rho²/q. Demeaning does not cancel the bias. Ratios have the positive variance guards in the claim. The usual measure-theoretic conditions are assumed.", "headline": "Supplement Section C, last paragraph (the explanation of why the uncorrected estimator is close to the true variance in the periodic design), corrected: the bias term is minus the sum of squared sample totals", "location": "Supplement p. A4 (staged/supplements/smj_suppl.pdf, PDF p. 4), Section C, last paragraph before the References; the design on supplement pp. A3–A4; Proposition 1 on main p. 234 (PDF p. 6)" }, "url": "https://pmaronow.github.io/llm-audit/works/P027/#P027-T2" }, { "declarations": [ { "claim_revision_id": "catalog:383405eb30d08dd3800e7f2aac61bb6ebca72bbed7c4e4228980b0df73a8e8f4", "conditional_on": [], "credit_reason": "Samplewise total equality and identical atom weights imply equal before/after variances; thus the exported expectations certify biases 8 and 9. It is an illustrative witness, not a refutation of an unspecified tendency.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P027.aggregation_counterexample_positive", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P027/P027_Audit.lean#L566", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:383405eb30d08dd3800e7f2aac61bb6ebca72bbed7c4e4228980b0df73a8e8f4", "conditional_on": [], "credit_reason": "The row supplies the second illustrative witness; it does not purport to negate a quantified distributional tendency.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P027.aggregation_counterexample_mixed", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P027/P027_Audit.lean#L582", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:383405eb30d08dd3800e7f2aac61bb6ebca72bbed7c4e4228980b0df73a8e8f4", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P027-T3", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P027-T3" }, "html": { "body": "

main p. 237, Section 3.1, with C2 from Section 3.2: the full probability tables, zero-pair lists and before/after estimator values for the two cluster witnesses. Totalling preserves the total and its variance but raises the conservative C2 biases from 8 to 9 and from 9/8 to 49/16, although the ordered zero-pair count falls from four to two. The usual measure-theoretic conditions are assumed.

", "headline": "Two examples where totalling raises bias, illustrating exceptions to the heuristic of Section 3.1", "location": "p. 237 (PDF p. 9), Section 3.1, paragraph after the proof of Corollary 1; Var^C2(t^)\\widehat{\\mathrm{Var}}_{C2}(\\hat t)VarC2​(t^) on the same page (Section 3.2)" }, "id": "P027-T3", "text": { "body": "main p. 237, Section 3.1, with C2 from Section 3.2: the full probability tables, zero-pair lists and before/after estimator values for the two cluster witnesses. Totalling preserves the total and its variance but raises the conservative C2 biases from 8 to 9 and from 9/8 to 49/16, although the ordered zero-pair count falls from four to two. The usual measure-theoretic conditions are assumed.", "headline": "Two examples where totalling raises bias, illustrating exceptions to the heuristic of Section 3.1", "location": "p. 237 (PDF p. 9), Section 3.1, paragraph after the proof of Corollary 1; $\\widehat{\\mathrm{Var}}_{C2}(\\hat t)$ on the same page (Section 3.2)" }, "url": "https://pmaronow.github.io/llm-audit/works/P027/#P027-T3" }, { "declarations": [ { "claim_revision_id": "catalog:0d91e415bb9a96fde0bae479ad8ab6cfcb7f0960fccc750d6df360f64a47bfb5", "conditional_on": [], "credit_reason": "Both ordered zero-pair directions are present and the factor 1/2 is correct. Positivity prevents diagonal zero-pair artifacts. 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It concludes the inclusion-probability identities, preservation of A and the actual variance, and samplewise preservation of the total and uncorrected estimator on positive-weight atoms. Under the claim's almost-sure co-inclusion premise, this support condition excludes only impossible samples introduced by allowing null atoms in the representation; equality at arbitrary such atoms is not a sampling assertion justified by that premise. 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Nonempty group sizes are already part of T4, and the exact lift represents arbitrary co-included groups on supported samples without assuming equal sizes or independence. The almost-sure version with possible null-atom disagreements is explicitly stated in P027/Wave1/AePartition.lean:17, so this coordinate representation loses none of the target's sampling assertions. 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The exact lift at line 635 represents every almost-sure cluster law on its support, with no equal-size or independence restriction; null-atom changes leave these biases unchanged. 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It permits arbitrary positive group sizes and arbitrary fixed outcomes, retaining the claimed sign and factors. Writing the original design as liftDesign changes only the coordinates of a co-included partition; every almost-sure partition has this form on the support, and the bias bound is invariant to selections on null atoms. 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main p. 237, Section 3.1: for every partition into nonempty almost-sure clusters, totalling preserves the HT total and the uncorrected estimator on every possible sample, and preserves variance and A. In the exponent-2 setting both A-star formulas and the exact ordered-pair-factor-1/2 bias change hold. Constant outcomes, strict increase for unequal linked sizes, the equal-size sufficient condition, the degree bound, the within-group squared-difference identity and both conservativeness formulas are included. The explicit examples reduce C2 bias from 20 to 0 and increase it from 8 to 9. The usual measure-theoretic conditions are assumed.

", "headline": "An optional precise qualification of what totalling preserves and when it does not increase bias", "location": "p. 237 (PDF p. 9), Section 3.1, paragraph after the proof of Corollary 1" }, "id": "P027-T4", "text": { "body": "main p. 237, Section 3.1: for every partition into nonempty almost-sure clusters, totalling preserves the HT total and the uncorrected estimator on every possible sample, and preserves variance and A. In the exponent-2 setting both A-star formulas and the exact ordered-pair-factor-1/2 bias change hold. Constant outcomes, strict increase for unequal linked sizes, the equal-size sufficient condition, the degree bound, the within-group squared-difference identity and both conservativeness formulas are included. The explicit examples reduce C2 bias from 20 to 0 and increase it from 8 to 9. The usual measure-theoretic conditions are assumed.", "headline": "An optional precise qualification of what totalling preserves and when it does not increase bias", "location": "p. 237 (PDF p. 9), Section 3.1, paragraph after the proof of Corollary 1" }, "url": "https://pmaronow.github.io/llm-audit/works/P027/#P027-T4" }, { "declarations": [ { "claim_revision_id": "catalog:9456bb68062ab109e37b35fc21ee63f2a3d2055993ca6f4451ee4ec0487ba421", "conditional_on": [], "credit_reason": "This row covers the upper-bound incompatibility part of C2 only; the population-level bound C1 is not assumed or established here, and the finite numerical check is not a full estimator refutation.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P027.a3_grouped_table_refutation", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P027/P027_Audit.lean#L772", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:9456bb68062ab109e37b35fc21ee63f2a3d2055993ca6f4451ee4ec0487ba421", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': \"P027/P027_Audit.lean:2949 certifies the centered population and the seven aggregate quantities of the T5(C3) witness in total units. The definition row21Fibre at line 786 is exactly the claimed population under k=2g_1+g_2+4j, and its four group totals are (-120,120,-120,120). The total-scale variance and three estimator expectations are 414720, 357120, 3766320, and 529920, whose division by the original 120^2 gives precisely the claim's mean-scale numbers and relative biases. The coordinate change and invertible scaling introduce no restriction or different estimator.\"}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P027.row21_exact_witness", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P027/P027_Audit.lean#L2797", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:9456bb68062ab109e37b35fc21ee63f2a3d2055993ca6f4451ee4ec0487ba421", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P027-T5", "decision": "restricted", "release_id": "sap-2026-11-08-assured-release", "target_id": "P027-T5" }, "html": { "body": "

Supplement Table A3 (p. A8), rows 21, 31, 41 and 51, interpreted with the aggregation instruction on main p. 237: four rounded table pairs violate the proposed grouped upper bound, and the 120-unit population has the displayed original and grouped estimator moments. Restricted: omits the universal population and finite-collection bounds, the four stronger numerical lower bounds for relative bias, the individual phase totals, and the displayed table columns other than true variance, C2 and q. Retains the scalar 120-unit witness and its mean-scale values using 120^2. No identification of the reported column with a particular implementation or with averages of grouped design expectations is asserted.

", "headline": "Supplement Table A3, columns Var^C2\\widehat{\\mathrm{Var}}_{C2}VarC2​ (rows 21, 31, 41, 51), read as the performance of the procedure the main text prescribes for clustered units (p. 237: ‘these units should be totaled into one larger unit before estimation’); the bound and the witness population are the validator's and are not in the paper", "location": "Supplement p. A8 (staged/supplements/smj_suppl.pdf, PDF p. 8), Table A3, rows 21, 31, 41, 51; supplement p. A4 (PDF p. 4), ‘while Var^C2(t^)\\widehat{\\mathrm{Var}}_{C2}(\\hat t)VarC2​(t^) is generally quite conservative’; main p. 237 (PDF p. 9), paragraph beginning ‘If any units k,lk, lk,l are in clusters’" }, "id": "P027-T5", "text": { "body": "Supplement Table A3 (p. A8), rows 21, 31, 41 and 51, interpreted with the aggregation instruction on main p. 237: four rounded table pairs violate the proposed grouped upper bound, and the 120-unit population has the displayed original and grouped estimator moments. Restricted: omits the universal population and finite-collection bounds, the four stronger numerical lower bounds for relative bias, the individual phase totals, and the displayed table columns other than true variance, C2 and q. Retains the scalar 120-unit witness and its mean-scale values using 120^2. No identification of the reported column with a particular implementation or with averages of grouped design expectations is asserted.", "headline": "Supplement Table A3, columns $\\widehat{\\mathrm{Var}}_{C2}$ (rows 21, 31, 41, 51), read as the performance of the procedure the main text prescribes for clustered units (p. 237: ‘these units should be totaled into one larger unit before estimation’); the bound and the witness population are the validator's and are not in the paper", "location": "Supplement p. A8 (staged/supplements/smj_suppl.pdf, PDF p. 8), Table A3, rows 21, 31, 41, 51; supplement p. A4 (PDF p. 4), ‘while $\\widehat{\\mathrm{Var}}_{C2}(\\hat t)$ is generally quite conservative’; main p. 237 (PDF p. 9), paragraph beginning ‘If any units $k, l$ are in clusters’" }, "url": "https://pmaronow.github.io/llm-audit/works/P027/#P027-T5" }, { "declarations": [ { "claim_revision_id": "catalog:d4a50c794d3267f5e109c2a7b1bae806417c16da55bb82888e0cebf2434dd3dd", "conditional_on": [], "credit_reason": "This is an explicit example for T6(vii), not a universal relative-bias identity. 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Its conjunction states the negative-square bias, actual variance, both C2 expectations, samplewise corrections and differences, deviation formula, conservativeness comparisons, and the claimed relative-bias bounds. The finite phase probabilities are exactly the sample distribution specified in the target; no Gaussian, equal-total, or independence assumption is added for these finite-population clauses. 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correction to Supplement Section C, pp. A3–A4 and Table A3 (p. A8): the original-unit and totalled periodic estimators have the displayed finite correction formulas, comparison, bounds and outcome-averaged ratio. Restricted: states the finite phase law, without a continuous uniform-start identification. Adds integrability of the model variance, grouped expected estimator and every squared group total. Omits separate assertions for the expectation of the unit-level correction and A-star, the expectation of the grouped correction, E_y(S)=qL and V's expression in terms of E_y(S), explicit mean-scale identities, separate grouped HT total/uncorrected-estimator and variance-preservation identities, and the original eight-unit representation of the realized-ratio example (its grouped totals are retained). No assertion about which implementation produced the table or figure is made.

", "headline": "Supplement Section C and Table A3 (the Var^C2\\widehat{\\mathrm{Var}}_{C2}VarC2​ results for the periodic design), corrected by the disclosure that the estimator was applied to the original units; formal content: the unit-level and the totalled Var^C2\\widehat{\\mathrm{Var}}_{C2}VarC2​ in the periodic design", "location": "Supplement p. A4 (staged/supplements/smj_suppl.pdf, PDF p. 4), discussion of Table A3 and Figure A3; Table A3 on supplement p. A8 (PDF p. 8); Figure A3 on supplement p. A10 (PDF p. 10); main p. 237 (PDF p. 9)" }, "id": "P027-T6", "text": { "body": "correction to Supplement Section C, pp. A3–A4 and Table A3 (p. A8): the original-unit and totalled periodic estimators have the displayed finite correction formulas, comparison, bounds and outcome-averaged ratio. Restricted: states the finite phase law, without a continuous uniform-start identification. Adds integrability of the model variance, grouped expected estimator and every squared group total. Omits separate assertions for the expectation of the unit-level correction and A-star, the expectation of the grouped correction, E_y(S)=qL and V's expression in terms of E_y(S), explicit mean-scale identities, separate grouped HT total/uncorrected-estimator and variance-preservation identities, and the original eight-unit representation of the realized-ratio example (its grouped totals are retained). No assertion about which implementation produced the table or figure is made.", "headline": "Supplement Section C and Table A3 (the $\\widehat{\\mathrm{Var}}_{C2}$ results for the periodic design), corrected by the disclosure that the estimator was applied to the original units; formal content: the unit-level and the totalled $\\widehat{\\mathrm{Var}}_{C2}$ in the periodic design", "location": "Supplement p. A4 (staged/supplements/smj_suppl.pdf, PDF p. 4), discussion of Table A3 and Figure A3; Table A3 on supplement p. A8 (PDF p. 8); Figure A3 on supplement p. A10 (PDF p. 10); main p. 237 (PDF p. 9)" }, "url": "https://pmaronow.github.io/llm-audit/works/P027/#P027-T6" }, { "declarations": [ { "claim_revision_id": "catalog:c4bef8cffe8923730a53a13b4e2a7fa09097bf88e74c8b6298b5e8ee4c90b03a", "conditional_on": [], "credit_reason": "Historical excluded association.", "credit_status": null, "fidelity_status_baseline": null, "lean_name": null, "manifest_use": "not_applicable", "named_axioms": [], "says": null, "source": null, "status": "not_applicable", "support_scope": "historical declaration", "used_in_solution": null } ], "formalization": { "claim_revision_id": "catalog:c4bef8cffe8923730a53a13b4e2a7fa09097bf88e74c8b6298b5e8ee4c90b03a", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P027-T9", "decision": "omitted", "release_id": "sap-2026-11-08-assured-release", "target_id": "P027-T9" }, "html": { "body": "

No current theorem is supplied for this target.

", "headline": "Supplement Table A3 (Var^C2\\widehat{\\mathrm{Var}}_{C2}VarC2​ columns) and the sentence ‘Var^C2(t^)\\widehat{\\mathrm{Var}}_{C2}(\\hat t)VarC2​(t^) is generally quite conservative’, as printed, with the instruction of main p. 237 to total clustered units", "location": "Supplement p. A8 (staged/supplements/smj_suppl.pdf, PDF p. 8), Table A3; supplement p. A4 (PDF p. 4), discussion of Table A3 and Figure A3; main p. 237 (PDF p. 9), paragraph beginning ‘If any units k,lk, lk,l are in clusters’" }, "id": "P027-T9", "text": { "body": "No current theorem is supplied for this target.", "headline": "Supplement Table A3 ($\\widehat{\\mathrm{Var}}_{C2}$ columns) and the sentence ‘$\\widehat{\\mathrm{Var}}_{C2}(\\hat t)$ is generally quite conservative’, as printed, with the instruction of main p. 237 to total clustered units", "location": "Supplement p. A8 (staged/supplements/smj_suppl.pdf, PDF p. 8), Table A3; supplement p. A4 (PDF p. 4), discussion of Table A3 and Figure A3; main p. 237 (PDF p. 9), paragraph beginning ‘If any units $k, l$ are in clusters’" }, "url": "https://pmaronow.github.io/llm-audit/works/P027/#P027-T9" }, { "declarations": [ { "claim_revision_id": "catalog:b77e027761a413ba0c98bed040b9cb3b1c1a6b3ff686ecb03141ffbf5d39b210", "conditional_on": [], "credit_reason": "This is only the advertised positivity component of the witness, not a standalone bias counterexample.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P027.random8_positive", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P027/P027_Audit.lean#L500", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:b77e027761a413ba0c98bed040b9cb3b1c1a6b3ff686ecb03141ffbf5d39b210", "conditional_on": [], "credit_reason": "These are actual samplewise values under the explicit probability law; expectation and mean-scale conclusions are provided by other credited rows.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P027.random8_sample_values", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P027/P027_Audit.lean#L595", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:b77e027761a413ba0c98bed040b9cb3b1c1a6b3ff686ecb03141ffbf5d39b210", "conditional_on": [], "credit_reason": "This row covers the fixed witness’s C3/C5 numerical conclusions, not certification of the full size/noise generator. 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Supplement p. A3, Section C and Table A2: the full positive-size witness (1,3,...) has the specified cumulative-size uniform-start law, marginal and joint probabilities, 48 ordered zero pairs, samplewise estimates, total/mean moments and downward bias. Its sizes have positive probability under the normalized offset negative-binomial generator and obey the PPS feasibility inequality; its outcomes have the additive residual representation for every beta. The entire stated open box has negative A, while C2 remains conservative. The usual measure-theoretic conditions are assumed. This witness refutes the inference from x_k >= 1 and the noisy model y_k = beta*x_k + epsilon_k to conservativity for every realized finite population. It does not refute the literal assertion for outcomes proportional to the positive size vector.

", "headline": "Supplement Section C: ‘Because yyy is proportional to xxx with x≥1x \\ge 1x≥1, Proposition 1 in the main text indicates that the Horvitz-Thompson variance estimator for μ^HT\\hat\\mu_{HT}μ^​HT​, (1/N2)Var^(t^)(1/N^2)\\widehat{\\mathrm{Var}}(\\hat t)(1/N2)Var(t^), provides a conservative approximation of the sampling variance of μ^HT\\hat\\mu_{HT}μ^​HT​’; the witness is the validator's construction and is not in the paper", "location": "Supplement p. A3 (staged/supplements/smj_suppl.pdf, PDF p. 3), Section C, paragraph beginning ‘Because yyy is proportional to xxx’; the outcome model yk=βxk+ϵky_k = \\beta x_k + \\epsilon_kyk​=βxk​+ϵk​ is displayed on the same page" }, "id": "P027-T10", "text": { "body": "Supplement p. A3, Section C and Table A2: the full positive-size witness (1,3,...) has the specified cumulative-size uniform-start law, marginal and joint probabilities, 48 ordered zero pairs, samplewise estimates, total/mean moments and downward bias. Its sizes have positive probability under the normalized offset negative-binomial generator and obey the PPS feasibility inequality; its outcomes have the additive residual representation for every beta. The entire stated open box has negative A, while C2 remains conservative. The usual measure-theoretic conditions are assumed. This witness refutes the inference from x_k >= 1 and the noisy model y_k = beta*x_k + epsilon_k to conservativity for every realized finite population. It does not refute the literal assertion for outcomes proportional to the positive size vector.", "headline": "Supplement Section C: ‘Because $y$ is proportional to $x$ with $x \\ge 1$, Proposition 1 in the main text indicates that the Horvitz-Thompson variance estimator for $\\hat\\mu_{HT}$, $(1/N^2)\\widehat{\\mathrm{Var}}(\\hat t)$, provides a conservative approximation of the sampling variance of $\\hat\\mu_{HT}$’; the witness is the validator's construction and is not in the paper", "location": "Supplement p. A3 (staged/supplements/smj_suppl.pdf, PDF p. 3), Section C, paragraph beginning ‘Because $y$ is proportional to $x$’; the outcome model $y_k = \\beta x_k + \\epsilon_k$ is displayed on the same page" }, "url": "https://pmaronow.github.io/llm-audit/works/P027/#P027-T10" }, { "declarations": [ { "claim_revision_id": "catalog:ccc7512e860ed1bbbfa8eca5f4b3ffc809f37b552740805d8702ebdae8fcb622", "conditional_on": [], "credit_reason": "Combined with the general averaged-bias identity this supplies 176β² and 396; it does not itself certify the negative realized-outcome example mentioned in T11(iv).", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P027.random8_outcome_coefficient", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P027/P027_Audit.lean#L941", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:ccc7512e860ed1bbbfa8eca5f4b3ffc809f37b552740805d8702ebdae8fcb622", "conditional_on": [], "credit_reason": "This auxiliary component is already supplied by an invoked aggregate result and is not separately invoked.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P027.expect_div_const", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P027/P027_Audit.lean#L1413", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:ccc7512e860ed1bbbfa8eca5f4b3ffc809f37b552740805d8702ebdae8fcb622", "conditional_on": [], "credit_reason": "This auxiliary component is already supplied by an invoked aggregate result and is not separately invoked.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P027.variance_div_const", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P027/P027_Audit.lean#L1418", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:ccc7512e860ed1bbbfa8eca5f4b3ffc809f37b552740805d8702ebdae8fcb622", "conditional_on": [], "credit_reason": "This auxiliary component is already supplied by an invoked aggregate result and is not separately invoked.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P027.mean_bias_identity", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P027/P027_Audit.lean#L1426", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:ccc7512e860ed1bbbfa8eca5f4b3ffc809f37b552740805d8702ebdae8fcb622", "conditional_on": [], "credit_reason": "Finite variances imply integrability of the individual errors and every product by Cauchy–Schwarz, so UncorrelatedNoise adds no moment restriction to the mathematical target. 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correction to Supplement Section C, p. A3: conditional on the positive size vector, mean-zero errors with finite second moments and zero cross moments give E[y_k y_l] = beta^2 x_k x_l for distinct units, E[A] = beta^2 sum_Z x_k x_l >= 0, and average mean-scale sampling bias E[A]/N^2 >= 0. The eight-unit coefficient is 176 (396 at beta = 3/2), whereas the displayed fixed outcome vector has A = -16 and mean-scale bias -1/4. For each fixed population conservativity is equivalent to A >= 0; outcomes all of one sign suffice. Thus the average guarantee supplies no guarantee for every realized population. The usual measure-theoretic conditions are assumed; the nonzero population size is presupposed by the population mean.

", "headline": "Supplement Section C, the sentence ‘Because yyy is proportional to xxx with x≥1x \\ge 1x≥1, Proposition 1 ... provides a conservative approximation’, corrected: nonnegative bias after averaging over the generated outcomes", "location": "Supplement p. A3 (staged/supplements/smj_suppl.pdf, PDF p. 3), Section C, paragraph beginning ‘Because yyy is proportional to xxx’" }, "id": "P027-T11", "text": { "body": "correction to Supplement Section C, p. A3: conditional on the positive size vector, mean-zero errors with finite second moments and zero cross moments give E[y_k y_l] = beta^2 x_k x_l for distinct units, E[A] = beta^2 sum_Z x_k x_l >= 0, and average mean-scale sampling bias E[A]/N^2 >= 0. The eight-unit coefficient is 176 (396 at beta = 3/2), whereas the displayed fixed outcome vector has A = -16 and mean-scale bias -1/4. For each fixed population conservativity is equivalent to A >= 0; outcomes all of one sign suffice. Thus the average guarantee supplies no guarantee for every realized population. The usual measure-theoretic conditions are assumed; the nonzero population size is presupposed by the population mean.", "headline": "Supplement Section C, the sentence ‘Because $y$ is proportional to $x$ with $x \\ge 1$, Proposition 1 ... provides a conservative approximation’, corrected: nonnegative bias after averaging over the generated outcomes", "location": "Supplement p. A3 (staged/supplements/smj_suppl.pdf, PDF p. 3), Section C, paragraph beginning ‘Because $y$ is proportional to $x$’" }, "url": "https://pmaronow.github.io/llm-audit/works/P027/#P027-T11" }, { "declarations": [ { "claim_revision_id": "catalog:a29441d73807d60cb8b2930adaf4f3045bee855e3214dbf46f57d2c8063a0cd5", "conditional_on": [], "credit_reason": "The genuine design uses finite sample atoms, which represent every distribution on subsets of this finite population. The record’s old review note says generated-law support is not credited without the sorry-bearing original nb_generated_witness; the separately credited Wave1.nb_generated_witness_proof now states the required normalization, moments and singleton positivity. 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witness concerning Supplement Section C, p. A3 and Table A2: the positive-probability offset-negative-binomial size vector gives a printed inclusion probability exceeding one, and each of the six configurations has such a positive-probability witness.

", "headline": "Supplement Section C, the display πk=xk∑l=1Nxlm\\pi_k = \\frac{x_k}{\\sum_{l=1}^{N} x_l}mπk​=∑l=1N​xl​xk​​m for the first-order inclusion probability, with the stated size law (negative binomial, mean 6, variance 16, offset 1); the witness is the validator's construction and is not in the paper", "location": "Supplement p. A3 (staged/supplements/smj_suppl.pdf, PDF p. 3), Section C: the sentence on the size variable and the display for πk\\pi_kπk​; the definition Ik∈{0,1}I_k \\in \\{0,1\\}Ik​∈{0,1}, E(Ik)=πk\\mathrm{E}(I_k) = \\pi_kE(Ik​)=πk​ is on main p. 233 (PDF p. 5)" }, "id": "P027-T12", "text": { "body": "witness concerning Supplement Section C, p. A3 and Table A2: the positive-probability offset-negative-binomial size vector gives a printed inclusion probability exceeding one, and each of the six configurations has such a positive-probability witness.", "headline": "Supplement Section C, the display $\\pi_k = \\frac{x_k}{\\sum_{l=1}^{N} x_l}m$ for the first-order inclusion probability, with the stated size law (negative binomial, mean 6, variance 16, offset 1); the witness is the validator's construction and is not in the paper", "location": "Supplement p. A3 (staged/supplements/smj_suppl.pdf, PDF p. 3), Section C: the sentence on the size variable and the display for $\\pi_k$; the definition $I_k \\in \\{0,1\\}$, $\\mathrm{E}(I_k) = \\pi_k$ is on main p. 233 (PDF p. 5)" }, "url": "https://pmaronow.github.io/llm-audit/works/P027/#P027-T12" }, { "declarations": [ { "claim_revision_id": "catalog:53cfe18c5e80629f55d279b4cf3f841ebea78c62e4e36b60d8e5ee7df7c856dd", "conditional_on": [], "credit_reason": "The statement matches the component credited to this row; no additional hypothesis, changed object, or omitted substantive conclusion was found.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P027.pps_feasibility_necessary", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P027/P027_Audit.lean#L833", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:53cfe18c5e80629f55d279b4cf3f841ebea78c62e4e36b60d8e5ee7df7c856dd", "conditional_on": [], "credit_reason": "The statement matches the component credited to this row; no additional hypothesis, changed object, or omitted substantive conclusion was found.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P027.capped_pps_exists", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P027/P027_Audit.lean#L846", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:53cfe18c5e80629f55d279b4cf3f841ebea78c62e4e36b60d8e5ee7df7c856dd", "conditional_on": [], "credit_reason": "The statement matches the component credited to this row; no additional hypothesis, changed object, or omitted substantive conclusion was found.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P027.redistributed_witness", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P027/P027_Audit.lean#L895", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:53cfe18c5e80629f55d279b4cf3f841ebea78c62e4e36b60d8e5ee7df7c856dd", "conditional_on": [], "credit_reason": "Scope is exactly the corrected given-probability construction T13(iii); this row is not credited as establishing the raw-size formula on infeasible vectors. 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correction to the PPS display in Supplement Section C, p. A3: for N >= 2, positive sizes and 0 < m < N, the printed probabilities require m max(x) <= sum(x). The cap-sum map is continuous, nondecreasing, starts at zero and eventually equals N; a positive cap multiplier gives sum m. Under feasibility, m/sum(x) retains the printed formula. Ordered intervals with these probabilities and a uniform start in [0,1) give exactly m distinct units, the prescribed marginals and the intersection-measure joint probabilities. For sizes (30,1,1,1,1,1,1,1), m = 2, redistribution gives (1,1/7,...,1/7): the same continuous design draws each {1,j} with probability 1/7, with the stated joints, whereas naive capping sums to 51/37 != 2. The formula-preserving recipe independently redraws full offset-NB vectors until feasibility; acceptance has positive probability, termination is almost sure and the output law is the original law restricted to feasibility and normalized. The alternative retains the original size law and uses the capped ordered design and its joints. The usual measure-theoretic conditions are assumed. No assertion about which recipe generated the unavailable simulation runs is made.

", "headline": "Supplement Section C, the display for the first-order inclusion probability under systematic PPS sampling, corrected: the feasibility condition mmax⁡kxk≤∑lxlm\\max_k x_k \\le \\sum_l x_lmmaxk​xk​≤∑l​xl​, or certainty-unit redistribution πk=min⁡(1,c xk)\\pi_k = \\min(1, c\\,x_k)πk​=min(1,cxk​) with ∑kπk=m\\sum_k\\pi_k = m∑k​πk​=m", "location": "Supplement p. A3 (staged/supplements/smj_suppl.pdf, PDF p. 3), Section C: the sentence on the size variable and the display for πk\\pi_kπk​" }, "id": "P027-T13", "text": { "body": "correction to the PPS display in Supplement Section C, p. A3: for N >= 2, positive sizes and 0 < m < N, the printed probabilities require m max(x) <= sum(x). The cap-sum map is continuous, nondecreasing, starts at zero and eventually equals N; a positive cap multiplier gives sum m. Under feasibility, m/sum(x) retains the printed formula. Ordered intervals with these probabilities and a uniform start in [0,1) give exactly m distinct units, the prescribed marginals and the intersection-measure joint probabilities. For sizes (30,1,1,1,1,1,1,1), m = 2, redistribution gives (1,1/7,...,1/7): the same continuous design draws each {1,j} with probability 1/7, with the stated joints, whereas naive capping sums to 51/37 != 2. The formula-preserving recipe independently redraws full offset-NB vectors until feasibility; acceptance has positive probability, termination is almost sure and the output law is the original law restricted to feasibility and normalized. The alternative retains the original size law and uses the capped ordered design and its joints. The usual measure-theoretic conditions are assumed. No assertion about which recipe generated the unavailable simulation runs is made.", "headline": "Supplement Section C, the display for the first-order inclusion probability under systematic PPS sampling, corrected: the feasibility condition $m\\max_k x_k \\le \\sum_l x_l$, or certainty-unit redistribution $\\pi_k = \\min(1, c\\,x_k)$ with $\\sum_k\\pi_k = m$", "location": "Supplement p. A3 (staged/supplements/smj_suppl.pdf, PDF p. 3), Section C: the sentence on the size variable and the display for $\\pi_k$" }, "url": "https://pmaronow.github.io/llm-audit/works/P027/#P027-T13" }, { "declarations": [ { "claim_revision_id": "catalog:9e59e39ff7c06807439a9348e47e3d63e05f692cec0db64819c0889204998980", "conditional_on": [], "credit_reason": "Formula arithmetic alone does not identify the actual model expectations required by C2-C3; those clauses are omitted.", "credit_status": "narrower", "fidelity_status_baseline": "narrower", "lean_name": "AronowAudit.P027.strata_model_numeric_witnesses", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P027/P027_Audit.lean#L1088", "status": "proved", "support_scope": "limited", "used_in_solution": false }, { "claim_revision_id": "catalog:9e59e39ff7c06807439a9348e47e3d63e05f692cec0db64819c0889204998980", "conditional_on": [], "credit_reason": "The target explicitly separates fixed-population C1 from model-averaged C2 and C3. This is an exact C1 witness and is not credited as a full Gaussian-model refutation or a mean-squared-error comparison.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P027.strata5_counterexample", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P027/P027_Audit.lean#L1993", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:9e59e39ff7c06807439a9348e47e3d63e05f692cec0db64819c0889204998980", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P027-T14", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P027-T14" }, "html": { "body": "

supplement Section B, p. A2 and Table A1, p. A5. At M=5 the two fixed populations have the stated true variances, all three expected estimates, biases, and reversed bias rankings. Independent standard Gaussian errors give the model-averaged biases 2/5 versus 1/10 at alpha=0 and 18/5 versus 41/10 at alpha=2, again reversing the ranking. At M=500 and alpha=0 the model averages of the conditional sampling variance, corrected estimate, and collapsed estimate are 499/1000, 499/500, and 1/2, with relative biases 1 and 1/499. All outcome expectations exist and the relative-bias denominator is positive. The rounded simulation rows are comparison context, rather than additional exact expectation identities.

", "headline": "Supplement Section B, last paragraph: ‘The (1/N2)Var^C2(t^)(1/N^2)\\widehat{\\mathrm{Var}}_{C2}(\\hat t)(1/N2)VarC2​(t^) estimator dominates V^CS(μ^HT)\\widehat V_{CS}(\\hat\\mu_{HT})VCS​(μ^​HT​) for smaller stratum sizes and the two perform similarly for larger stratum sizes’; the witnesses are the validator's and are not in the paper", "location": "Supplement p. A2 (staged/supplements/smj_suppl.pdf, PDF p. 2), Section B, last paragraph; Table A1 on supplement p. A5 (PDF p. 5)" }, "id": "P027-T14", "text": { "body": "supplement Section B, p. A2 and Table A1, p. A5. At M=5 the two fixed populations have the stated true variances, all three expected estimates, biases, and reversed bias rankings. Independent standard Gaussian errors give the model-averaged biases 2/5 versus 1/10 at alpha=0 and 18/5 versus 41/10 at alpha=2, again reversing the ranking. 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supplement Section B, pp. A1–A2. The sample formulas, fixed-population variance, model expectations, nonnegative biases, exact parameter-qualified ranking, and finite ratios and limits give the corrected comparison.

", "headline": "Supplement Section B, last paragraph, the comparison of (1/N2)Var^C2(t^)(1/N^2)\\widehat{\\mathrm{Var}}_{C2}(\\hat t)(1/N2)VarC2​(t^) with V^CS(μ^HT)\\widehat V_{CS}(\\hat\\mu_{HT})VCS​(μ^​HT​), corrected: the bias ranking by MMM and α\\alphaα", "location": "Supplement p. A2 (staged/supplements/smj_suppl.pdf, PDF p. 2), Section B, last paragraph; the design and estimators on supplement pp. A1–A2" }, "id": "P027-T15", "text": { "body": "supplement Section B, pp. A1–A2. The sample formulas, fixed-population variance, model expectations, nonnegative biases, exact parameter-qualified ranking, and finite ratios and limits give the corrected comparison.", "headline": "Supplement Section B, last paragraph, the comparison of $(1/N^2)\\widehat{\\mathrm{Var}}_{C2}(\\hat t)$ with $\\widehat V_{CS}(\\hat\\mu_{HT})$, corrected: the bias ranking by $M$ and $\\alpha$", "location": "Supplement p. A2 (staged/supplements/smj_suppl.pdf, PDF p. 2), Section B, last paragraph; the design and estimators on supplement pp. A1–A2" }, "url": "https://pmaronow.github.io/llm-audit/works/P027/#P027-T15" }, { "declarations": [ { "claim_revision_id": "catalog:0e21d0fa51dca9ef6c157f11bee5e542796d1002f2d364733220fb80d24b31f2", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': \"AronowAudit/P027/P027_Audit.lean:1143 computes the claimed means, variances, covariance, correlation 1/5 and printed ratio 1/3 under the uniform product law defined at AronowAudit/P027/P027_Audit.lean:1128. T16's own explicit “Model” paragraph specifies centered unit-variance errors, independence, and the cross moments used; it does not specify Gaussian marginals in that restatement of the validator's witness. The symmetric-sign construction satisfies those requirements, so the Gaussian source-model display in definitions_needed does not justify rejecting this explicitly stated moment-model component. This verdict credits the numerical pair-correlation witness, not a separately asserted Gaussian-law instance.\"}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P027.icc_counterexample", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P027/P027_Audit.lean#L1123", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": false } ], "formalization": { "claim_revision_id": "catalog:0e21d0fa51dca9ef6c157f11bee5e542796d1002f2d364733220fb80d24b31f2", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P027-T16", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P027-T16" }, "html": { "body": "

supplement Section B, p. A2, with labels on pp. A5–A6. For an equally likely common stratum and independent standard Gaussian errors, alpha=1/2 gives zero outcome means, variances 5/4, covariance 1/4, and Pearson correlation 1/5. The printed ratio is 1/3 and exceeds that correlation by 2/15. The two fixed means average to zero; their divisor-1 variance is 1/2, giving the printed variance ratio, while their divisor-2 variance is 1/4, equal to the variance of the random stratum effect. Conditional on either stratum the outcome variances are one and their covariance and correlation are zero. All variables used in these correlations have finite second moments, and each conditioning event has probability 1/2. This compares the printed parameter with ordinary Pearson correlation and does not identify the authors' intended convention.

", "headline": "Supplement Section B: ‘the intra-class correlation over strata is given by, ρICC=α2α2+1/2\\rho_{ICC} = \\frac{\\alpha^2}{\\alpha^2 + 1/2}ρICC​=α2+1/2α2​’; the witness is the validator's and is not in the paper", "location": "Supplement p. A2 (staged/supplements/smj_suppl.pdf, PDF p. 2), Section B, first lines of the page; the labels of Table A1 (supplement p. A5) and Figure A1 (supplement p. A6) use the parameter" }, "id": "P027-T16", "text": { "body": "supplement Section B, p. A2, with labels on pp. A5–A6. For an equally likely common stratum and independent standard Gaussian errors, alpha=1/2 gives zero outcome means, variances 5/4, covariance 1/4, and Pearson correlation 1/5. The printed ratio is 1/3 and exceeds that correlation by 2/15. The two fixed means average to zero; their divisor-1 variance is 1/2, giving the printed variance ratio, while their divisor-2 variance is 1/4, equal to the variance of the random stratum effect. Conditional on either stratum the outcome variances are one and their covariance and correlation are zero. All variables used in these correlations have finite second moments, and each conditioning event has probability 1/2. This compares the printed parameter with ordinary Pearson correlation and does not identify the authors' intended convention.", "headline": "Supplement Section B: ‘the intra-class correlation over strata is given by, $\\rho_{ICC} = \\frac{\\alpha^2}{\\alpha^2 + 1/2}$’; the witness is the validator's and is not in the paper", "location": "Supplement p. A2 (staged/supplements/smj_suppl.pdf, PDF p. 2), Section B, first lines of the page; the labels of Table A1 (supplement p. A5) and Figure A1 (supplement p. A6) use the parameter" }, "url": "https://pmaronow.github.io/llm-audit/works/P027/#P027-T16" }, { "declarations": [ { "claim_revision_id": "catalog:af3ec7f013580f6961249db8e642f28da4376c6746484138810eb614655ada2c", "conditional_on": [], "credit_reason": "q is the target-defined heterogeneity(alpha), so parameterizing the algebra by alpha does not restrict its cases; every 0<=q<1 is represented by alpha=sqrt(q/(2*(1-q))).", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P027.heterogeneity_conversion", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P027/P027_Audit.lean#L1140", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:af3ec7f013580f6961249db8e642f28da4376c6746484138810eb614655ada2c", "conditional_on": [], "credit_reason": "Fidelity sweep: ok", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P027.heterogeneity_conversion_monotone", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P027/P027_Audit.lean#L1182", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:af3ec7f013580f6961249db8e642f28da4376c6746484138810eb614655ada2c", "conditional_on": [], "credit_reason": "Fidelity sweep: ok", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P027.printed_icc_labels", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P027/P027_Audit.lean#L1189", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:af3ec7f013580f6961249db8e642f28da4376c6746484138810eb614655ada2c", "conditional_on": [], "credit_reason": "Fidelity sweep: ok", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P027.heterogeneity_conversion_limit", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P027/P027_Audit.lean#L1407", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:af3ec7f013580f6961249db8e642f28da4376c6746484138810eb614655ada2c", "conditional_on": [], "credit_reason": "Fidelity sweep: ok", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P027.stratum_means_variance", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P027/P027_Audit.lean#L2057", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:af3ec7f013580f6961249db8e642f28da4376c6746484138810eb614655ada2c", "conditional_on": [], "credit_reason": "The raw-moment correlation is the ordinary correlation because both zero means are included in the conclusion; all its variance denominators are at least 1.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P027.same_stratum_correlation", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P027/P027_Audit.lean#L2274", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:af3ec7f013580f6961249db8e642f28da4376c6746484138810eb614655ada2c", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P027-T17", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P027-T17" }, "html": { "body": "

supplement Section B, p. A2, Table A1 and Figure A1, pp. A5–A6. The printed heterogeneity parameter uses divisor-1 stratum-mean variance, and its conversion to ordinary same-stratum correlation has the stated comparison, monotonicity, limit, and labels.

", "headline": "Supplement Section B, the display for ρICC\\rho_{ICC}ρICC​, corrected: the printed ratio is a variance-ratio parameter with the divisor-1 variance of the two stratum means; the same-stratum pair correlation is α2/(α2+1)\\alpha^2/(\\alpha^2 + 1)α2/(α2+1)", "location": "Supplement p. A2 (staged/supplements/smj_suppl.pdf, PDF p. 2), Section B, first lines of the page; Table A1 (supplement p. A5) and Figure A1 (supplement p. A6)" }, "id": "P027-T17", "text": { "body": "supplement Section B, p. A2, Table A1 and Figure A1, pp. A5–A6. The printed heterogeneity parameter uses divisor-1 stratum-mean variance, and its conversion to ordinary same-stratum correlation has the stated comparison, monotonicity, limit, and labels.", "headline": "Supplement Section B, the display for $\\rho_{ICC}$, corrected: the printed ratio is a variance-ratio parameter with the divisor-1 variance of the two stratum means; the same-stratum pair correlation is $\\alpha^2/(\\alpha^2 + 1)$", "location": "Supplement p. A2 (staged/supplements/smj_suppl.pdf, PDF p. 2), Section B, first lines of the page; Table A1 (supplement p. A5) and Figure A1 (supplement p. A6)" }, "url": "https://pmaronow.github.io/llm-audit/works/P027/#P027-T17" }, { "declarations": [ { "claim_revision_id": "catalog:0b5d8c7594356cecad30126adecfbeb955f4f70d4efe7831dc91f04a01df547a", "conditional_on": [], "credit_reason": "The first-order inclusion formula is supplied by strata_inclusion_correction.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P027.strata_pi_value", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P027/P027_Audit.lean#L1650", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:0b5d8c7594356cecad30126adecfbeb955f4f70d4efe7831dc91f04a01df547a", "conditional_on": [], "credit_reason": "The diagonal branch precedes the same-stratum branch, exactly implementing the correction. The ε-index repair is a definition-level typo and is not a separate probabilistic theorem claimed by this row.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P027.strata_joint_value", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P027/P027_Audit.lean#L1659", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:0b5d8c7594356cecad30126adecfbeb955f4f70d4efe7831dc91f04a01df547a", "conditional_on": [], "credit_reason": "Independent uniform stratum draws are explicitly part of the corrected target’s setting. The statement proves the corrected formula, rather than the erroneous printed formula assigning zero to the diagonal.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P027.strata_inclusion_correction", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P027/P027_Audit.lean#L1672", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:0b5d8c7594356cecad30126adecfbeb955f4f70d4efe7831dc91f04a01df547a", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P027-T18", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P027-T18" }, "html": { "body": "

supplement Section B, pp. A1–A2; main p. 233. Independent uniform stratum draws have diagonal inclusion 1/M, distinct same-stratum joint inclusion zero, cross-stratum inclusion 1/M squared, and the stated covariances. The separate outcome-index correction is the definition stratumOutcomes.

", "headline": "Supplement Section B, the display for the joint inclusion probabilities πks,lt\\pi_{ks,lt}πks,lt​ (the restriction to distinct units is missing) and the display yks=αs+ϵisy_{ks} = \\alpha_s + \\epsilon_{is}yks​=αs​+ϵis​ (index iii for kkk), corrected", "location": "Supplement p. A1 (staged/supplements/smj_suppl.pdf, PDF p. 1), Section B, the last two displays of the page; continued on supplement p. A2 (PDF p. 2); the convention πkk=πk\\pi_{kk} = \\pi_kπkk​=πk​ is on main p. 233 (PDF p. 5)" }, "id": "P027-T18", "text": { "body": "supplement Section B, pp. A1–A2; main p. 233. Independent uniform stratum draws have diagonal inclusion 1/M, distinct same-stratum joint inclusion zero, cross-stratum inclusion 1/M squared, and the stated covariances. The separate outcome-index correction is the definition stratumOutcomes.", "headline": "Supplement Section B, the display for the joint inclusion probabilities $\\pi_{ks,lt}$ (the restriction to distinct units is missing) and the display $y_{ks} = \\alpha_s + \\epsilon_{is}$ (index $i$ for $k$), corrected", "location": "Supplement p. A1 (staged/supplements/smj_suppl.pdf, PDF p. 1), Section B, the last two displays of the page; continued on supplement p. A2 (PDF p. 2); the convention $\\pi_{kk} = \\pi_k$ is on main p. 233 (PDF p. 5)" }, "url": "https://pmaronow.github.io/llm-audit/works/P027/#P027-T18" }, { "declarations": [ { "claim_revision_id": "CT-02:P027-T19:accepted", "conditional_on": [], "credit_reason": "The PhaseDecomposition structure encodes component membership and reconstruction only; existence, uniqueness, dimensions, orthogonality and the contrast formula are conclusions rather than assumptions.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P027.Wave1.contrast_deterministic_correction_proof", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P027/Wave1/Gap/GLP027P016P031B2.lean#L520", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "CT-02:P027-T19:accepted", "conditional_on": [], "credit_reason": "{'source': 'adjudication+second review', 'adjudicator': \"AronowAudit/P027/Wave1/Gap/GLP027P016P031B2.lean:796 assumes 0 < sigma2, whereas T19(iv) states its Gaussian representation and “E(T)=a(N-q)>0” without an explicit positive-variance premise. The zero-variance Gaussian law is admitted by GaussianNoise at AronowAudit/P027/P027_Audit.lean:1355; then the contrast is almost surely zero and the target's strict positivity is false. This is a minor correction to the target's domain because the excluded case is deterministic and mathematically trivial, rather than the major loss alleged by the reviewer. The separate omission of an explicit AEMeasurable conjunct for the existential witnesses is a minor zero-degree chi-square convention issue, not an additional material restriction.\", 'suggested_fix': 'State sigma^2>0 explicitly in T19(iv); if covering sigma^2=0, state T=0 almost surely and E(T)=0 separately. Exporting AEMeasurable for each chi-square witness would also remove the minor chi-square-zero convention ambiguity.', 'second_review': 'T19(iv) asserts E(T) = a(N-q) > 0 for independent N(0,σ²) errors without a positive-variance qualification, whereas P027/Wave1/Gap/GLP027P016P031B2.lean:798 requires 0 < σ². GaussianNoise at P027/P027_Audit.lean:1355 admits σ² = 0, since Mathlib/Probability/Distributions/Gaussian/Real.lean:229 identifies that Gaussian with a point mass; then the contrast is zero almost surely and E(T) = 0. The target correction therefore needs σ² > 0 for its strict-positivity assertion, a minor exclusion of a deterministic boundary case. The additional omitted AEMeasurable witness conjunct at P027/Wave1/Gap/GLP027P016P031B2.lean:804 is also only a minor zero-degree convention issue: chiSquaredLaw 0 is Dirac (P027/P027_Audit.lean:1362), and Measure.map has an arbitrary-Dirac fallback (Mathlib/MeasureTheory/Measure/Map.lean:99).', 'severity': None}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P027.Wave1.contrast_gaussian_law_proof", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P027/Wave1/Gap/GLP027P016P031B2.lean#L794", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "CT-02:P027-T19:accepted", "conditional_on": [], "credit_reason": "{'source': 'adjudication+second review', 'adjudicator': \"AronowAudit/P027/Wave1/Gap/GP027P016P031B1b.lean:626 requires 0 < sigma2; T19(v) does not explicitly include that domain restriction for its inherited Gaussian noise law. The only additional excluded case is sigma2=0, when every contrast and its average are almost surely zero, so the probability of Tbar<=-ell is zero for ell>0. The target's displayed exponential formula divides by k=ac, which is zero at that boundary, so this is the minor domain/convention issue described by the reviewer, with no loss of a nontrivial negative-tail case.\", 'suggested_fix': 'Specify sigma^2>0 (equivalently k>0) for the displayed T19(v) formulas, and, if desired, append the sigma^2=0 case with negative-tail probability zero.', 'second_review': 'P027/Wave1/Gap/GP027P016P031B1b.lean:626 exports the dependent-population tail bounds under 0 < σ² at line 629, with no independence assumption across populations. T19(v) inherits the Gaussian law without stating this qualification, although its displayed exp(-ℓ/(4k)) and floor(ℓ/(4k)) require a convention when k = ac = 0. At the σ² = 0 boundary allowed by GaussianNoise (P027/P027_Audit.lean:1355), the signal cancels, every contrast is zero almost surely, and the negative-tail probability is zero for ℓ > 0. Qualify the target display by σ² > 0, optionally stating this trivial zero-noise case separately; this is a minor claim-text domain repair.', 'severity': None}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P027.Wave1.contrast_gaussian_tail_proof", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P027/Wave1/Gap/GP027P016P031B1b.lean#L626", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "CT-02:P027-T19:accepted", "conditional_on": [], "credit_reason": "Fidelity sweep: ok", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P027.a3_high_contrast_arithmetic", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P027/P027_Audit.lean#L1336", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "CT-02:P027-T19:accepted", "conditional_on": [], "credit_reason": "Metadata inconsistency: T19 ends with a categorical claim that the printed means are not both exact-design averages, whereas its numbered clause (vi) and the record target note assert only probability bounds. This row is faithful to (vi), not a proof of that categorical assertion.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P027.Wave1.a3_high_rows_probability_bound_proof", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P027/Wave1/Gap/GP027P016P031B1b.lean#L659", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "CT-02:P027-T19:accepted", "conditional_on": [], "credit_reason": "The credited component is explicitly the numerical evaluation of a formula; it is not miscredited as identifying unavailable realized simulation outputs.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P027.periodic_model_prediction_values", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P027/P027_Audit.lean#L1398", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "CT-02:P027-T19:accepted", "conditional_on": [], "credit_reason": "The target setting itself specifies the phase samples and probabilities, so a separate continuous random-start equivalence theorem is not a missing premise of this scoped statement. The record review note has stale unproved language, contrary to the task proof-status premise; proofs were not inspected.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P027.periodic_mean_identities", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P027/P027_Audit.lean#L2938", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "CT-02:P027-T19:accepted", "conditional_on": [], "credit_reason": "This moment calculation is incorporated in periodic_model_moments; its full model identities are applied directly.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P027.centered_unit_second_moment", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P027/P027_Audit.lean#L3286", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "CT-02:P027-T19:accepted", "conditional_on": [], "credit_reason": "This moment calculation is incorporated in periodic_model_moments; its full model identities are applied directly.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P027.phase_centered_second_moment", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P027/P027_Audit.lean#L3395", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "CT-02:P027-T19:accepted", "conditional_on": [], "credit_reason": "This moment calculation is incorporated in periodic_model_moments; its full model identities are applied directly.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P027.phase_inverse_sum", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P027/P027_Audit.lean#L3419", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "CT-02:P027-T19:accepted", "conditional_on": [], "credit_reason": "The auxiliary identity also permits m=1; its specialization to the target m>=2 is unchanged, and its moment assumptions force sigma2>=0 on the nonempty unit type.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P027.periodic_model_bias", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P027/P027_Audit.lean#L3432", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "CT-02:P027-T19:accepted", "conditional_on": [], "credit_reason": "{'source': 'adjudication+second review', 'adjudicator': 'AronowAudit/P027/P027_Audit.lean:3666 makes the relative-bias identities conditional on 0 < L, unlike the unqualified ratio in T19(ii). With positive m and q, L=16*m^2*beta^2+m*sigma2*(1-1/q) vanishes only when beta=sigma2=0; in that model both averaged variance and averaged HT estimate are zero. The claimed ratio is then undefined in ordinary mathematics, or zero under Lean total division, rather than -5/(9*q). The flag therefore identifies a real but minor missing domain condition in the target, since only the mathematically trivial zero model is excluded; the other moment identities and the change from total to mean scale lose no cases.', 'suggested_fix': 'Keep the moment identities unconditional and qualify the relative-bias assertions in T19(ii) by E(V)>0, equivalently L>0; describe beta=sigma^2=0 separately as the zero-variance model.', 'second_review': 'P027/P027_Audit.lean:3666 exports the actual model moments and makes the relative-bias identities conditional on L > 0 at P027/P027_Audit.lean:3689. T19(ii) states (E(H)-E(V))/E(V) = -5/(9q) without that condition, although β = σ² = 0 with identically zero errors satisfies NoiseMoments (P027/P027_Audit.lean:934) and gives E(V) = E(H) = L = 0. The quotient is then undefined in ordinary mathematics, or zero under Lean total division, so the target needs E(V) > 0, equivalently L > 0; only the trivial zero model is excluded. The remaining moment identities retain that case, and the total-to-mean scaling follows from P027/P027_Audit.lean:1534 and P027/P027_Audit.lean:1539.', 'severity': None}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P027.periodic_model_moments", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P027/P027_Audit.lean#L3514", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "CT-02:P027-T19:accepted", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P027-T19", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P027-T19" }, "html": { "body": "

supplement Sections A and C, pp. A1, A3–A4, and Table A3, p. A8, rows 30,40,50,60. Exact periodic-design expectations and model moments, signal-canceling diagonalization, Gaussian chi-square laws and tail bounds give the stated tiny probabilities for the printed means. Ratios require positive expected variance, and Gaussian law and tail assertions require positive noise variance, exactly as in the accepted canonical claim; no deterministic exclusion is asserted.

", "headline": "Supplement Table A3, rows 30, 40, 50, 60 (the Var^HT\\widehat{\\mathrm{Var}}_{HT}VarHT​ and Var^C2\\widehat{\\mathrm{Var}}_{C2}VarC2​ means at ρ=0.99\\rho = 0.99ρ=0.99, N≥120N \\ge 120N≥120), with Section A's statement that these expected values ‘were computed exactly using expressions from Propositions 1 and 2’: the formal content of the finding (exact expectations in the stated design, their model moments, and a signal-free contrast); no corrected table values are available", "location": "Supplement p. A8 (staged/supplements/smj_suppl.pdf, PDF p. 8), Table A3, rows 30, 40, 50, 60; supplement p. A1 (PDF p. 1), Section A, second sentence; the design on supplement pp. A3–A4 (PDF pp. 3–4)" }, "id": "P027-T19", "text": { "body": "supplement Sections A and C, pp. A1, A3–A4, and Table A3, p. A8, rows 30,40,50,60. 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Proposition 1 and its following paragraph, p. 234, Section 2.2. The expected HT variance estimate equals the true total variance plus the ordered zero-pair bias A; common-sign outcomes give nonnegative bias, and examples realize either sign.

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Proposition 2 and proof, pp. 235–236, Section 3. For positive conjugate exponents the correction is unbiased for A-star, which bounds both signs of A, giving the exact corrected expectation and conservativeness for arbitrary outcomes.

", "headline": "Proposition 2", "location": "pp. 235–236 (PDF pp. 7–8), Section 3.1, with its proof" }, "id": "P027-T21", "text": { "body": "Proposition 2 and proof, pp. 235–236, Section 3. For positive conjugate exponents the correction is unbiased for A-star, which bounds both signs of A, giving the exact corrected expectation and conservativeness for arbitrary outcomes.", "headline": "Proposition 2", "location": "pp. 235–236 (PDF pp. 7–8), Section 3.1, with its proof" }, "url": "https://pmaronow.github.io/llm-audit/works/P027/#P027-T21" }, { "declarations": [ { "claim_revision_id": "catalog:5f9eb083126571bc274f61d86b0065d6e918715a72001c66d701d66f08a4dd81", "conditional_on": [], "credit_reason": "The supplied printed text already includes the inherited positive conjugate-exponent hypotheses and states sufficiency only; making them explicit changes neither cases nor meaning. The public description matches the statement.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P027.corollary1", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P027/P027_Audit.lean#L324", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:5f9eb083126571bc274f61d86b0065d6e918715a72001c66d701d66f08a4dd81", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P027-T22", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P027-T22" }, "html": { "body": "

Corollary 1 and proof, pp. 236–237, Section 3.1. Equality of the conjugate powers and opposite-sign equality on every zero-inclusion pair suffice for the corrected estimator to be unbiased.

", "headline": "Corollary 1", "location": "pp. 236–237 (PDF pp. 8–9), Section 3.1, with its proof" }, "id": "P027-T22", "text": { "body": "Corollary 1 and proof, pp. 236–237, Section 3.1. Equality of the conjugate powers and opposite-sign equality on every zero-inclusion pair suffice for the corrected estimator to be unbiased.", "headline": "Corollary 1", "location": "pp. 236–237 (PDF pp. 8–9), Section 3.1, with its proof" }, "url": "https://pmaronow.github.io/llm-audit/works/P027/#P027-T22" }, { "declarations": [ { "claim_revision_id": "catalog:56249e1a30cdc97232b509e055765b9f571595b61ba51d0eb5ea639b934dc638", "conditional_on": [], "credit_reason": "Alternative bias or expectation identity is unnecessary for the printed conservativeness conclusion.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P027.c2_bias_identity", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P027/P027_Audit.lean#L361", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:56249e1a30cdc97232b509e055765b9f571595b61ba51d0eb5ea639b934dc638", "conditional_on": [], "credit_reason": "This is the principal theorem establishing the whole target: conservativeness in design expectation, with arbitrary fixed real outcomes. The printed Proposition 1 reference is a citation misprint; the corrected reference does not change the asserted mathematical inequality. The record gives no reason to change the claim-stage verdict.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P027.corollary2", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P027/P027_Audit.lean#L378", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:56249e1a30cdc97232b509e055765b9f571595b61ba51d0eb5ea639b934dc638", "conditional_on": [], "credit_reason": "The extension beyond positive inclusion probabilities is harmless: setting both exponent arrays to two gives exactly the same function on every atom, with identical zero-denominator conventions. On the paper’s domain all denominators are positive. The record discloses that supporting identities can be more general, but that disclosure does not change the broader-than-claim classification saved at claim stage.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P027.c2_is_young_two", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P027/P027_Audit.lean#L388", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:56249e1a30cdc97232b509e055765b9f571595b61ba51d0eb5ea639b934dc638", "conditional_on": [], "credit_reason": "Alternative bias or expectation identity is unnecessary for the printed conservativeness conclusion.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P027.c2_expectation", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P027/P027_Audit.lean#L2072", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false }, { "claim_revision_id": "catalog:56249e1a30cdc97232b509e055765b9f571595b61ba51d0eb5ea639b934dc638", "conditional_on": [], "credit_reason": "Alternative correction rewriting is unnecessary for the displayed estimator and its conservativeness.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P027.quadratic_correction_degree", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P027/P027_Audit.lean#L2444", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false } ], "formalization": { "claim_revision_id": "catalog:56249e1a30cdc97232b509e055765b9f571595b61ba51d0eb5ea639b934dc638", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P027-T23", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P027-T23" }, "html": { "body": "

Corollary 2, p. 237, Section 3.2. The displayed quadratic estimator is the exponent-two specialization and is conservative.

", "headline": "Corollary 2", "location": "p. 237 (PDF p. 9), Section 3.2, with its one-line proof" }, "id": "P027-T23", "text": { "body": "Corollary 2, p. 237, Section 3.2. The displayed quadratic estimator is the exponent-two specialization and is conservative.", "headline": "Corollary 2", "location": "p. 237 (PDF p. 9), Section 3.2, with its one-line proof" }, "url": "https://pmaronow.github.io/llm-audit/works/P027/#P027-T23" }, { "declarations": [ { "claim_revision_id": "catalog:c37dff474fedaeaecb87c743c87c7722b8a68529f18597fa31719d946410cad7", "conditional_on": [], "credit_reason": "Ordered-pair symmetry is used inside corollary3; its complete samplewise conclusion is applied directly.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P027.zero_pair_sum_swap", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P027/P027_Audit.lean#L396", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:c37dff474fedaeaecb87c743c87c7722b8a68529f18597fa31719d946410cad7", "conditional_on": [], "credit_reason": "Attainment at the admissible conjugate pair (2,2), together with comparison against every constant conjugate pair, is exactly the printed samplewise minimum; pair-dependent exponents are outside the printed claim.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P027.corollary3", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P027/P027_Audit.lean#L407", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:c37dff474fedaeaecb87c743c87c7722b8a68529f18597fa31719d946410cad7", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P027-T24", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P027-T24" }, "html": { "body": "

Corollary 3 and proof, pp. 237–238, Section 3.2. On every sample the quadratic estimator attains the minimum among all positive constant conjugate-exponent corrected estimators, with equality at the pair (2,2).

", "headline": "Corollary 3", "location": "pp. 237–238 (PDF pp. 9–10), Section 3.2, with its proof" }, "id": "P027-T24", "text": { "body": "Corollary 3 and proof, pp. 237–238, Section 3.2. On every sample the quadratic estimator attains the minimum among all positive constant conjugate-exponent corrected estimators, with equality at the pair (2,2).", "headline": "Corollary 3", "location": "pp. 237–238 (PDF pp. 9–10), Section 3.2, with its proof" }, "url": "https://pmaronow.github.io/llm-audit/works/P027/#P027-T24" } ] }, "missing_sources": [], "reproduction": { "agreement": "agrees_in_part", "status": "selected_recalculation", "targets": [ { "agreement": "to_rounding", "location": "Supplement p. A5 (supplement PDF p. 5)", "method": "I recalculated nominal and relative biases from the reported means and true variances for selected rows. The cells are jointly compatible with two-decimal rounding; this checks internal arithmetic, not reproduction of the original simulation.", "status": "recalculated", "target": "Table A1" }, { "agreement": "agrees_in_part", "location": "Supplement p. A6 (supplement PDF p. 6)", "method": "I inspected the figure against the table trends and visible points, which agree. I did not regenerate the figure or compare independently computed coordinates. Its $\\rho_{ICC}$ labels describe a variance ratio rather than the ordinary same-stratum correlation (P027-A001).", "status": "inspected_only", "target": "Figure A1" }, { "agreement": "to_rounding", "location": "Supplement p. A7 (supplement PDF p. 7)", "method": "I recalculated nominal and relative biases from the reported means and true variances for selected rows. The cells are jointly compatible with two-decimal rounding; this checks internal arithmetic, not reproduction of the original simulation.", "status": "recalculated", "target": "Table A2" }, { "agreement": "agrees_in_part", "location": "Supplement p. A9 (supplement PDF p. 9)", "method": "I inspected the figure against the table trends and visible points, which agree. I did not regenerate the figure or compare independently computed coordinates. Figure A2’s plotting range is qualified in P027-E008.", "status": "inspected_only", "target": "Figure A2" }, { "agreement": "agrees_in_part", "location": "Supplement p. A8 (supplement PDF p. 8)", "method": "I recalculated selected bias columns from the printed means and true variances; their internal arithmetic is compatible with rounding. The HT and C2 means in four high-correlation rows are inconsistent with the described simulation (P027-N004), and the C2 columns describe the original units rather than the recommended totalled units (P027-N005). The original simulations could not be rerun.", "status": "recalculated", "target": "Table A3" }, { "agreement": "agrees_in_part", "location": "Supplement p. A10 (supplement PDF p. 10)", "method": "I inspected the figure against the table trends and visible points, which agree. I did not regenerate the figure or compare independently computed coordinates. Its legend calls the uncorrected HT series conservative (P027-E007), and its $C2$ series is for the original units, without the recommended totalling (P027-N005).", "status": "inspected_only", "target": "Figure A3" } ] }, "summary": "The paper studies variance estimation for the Horvitz--Thompson estimator of a population total when some pairs of units are never sampled together. It derives the exact bias of the standard variance estimator and a correction that is conservative in expectation; a supplement reports simulations. 9 entries are classified as errors (1 Moderate, 8 Minor), and 5 are clarity notes; 1 allegation was not confirmed and 1 item is undetermined. The results for $\\widehat{\\operatorname{Var}}_{C2}$ in the periodic simulation are for the estimator applied to the original units, without the totalling that the main text recommends (\\fid{P027-N005}). In the supplement's centered periodic design, the elements of the bias term cancel only in part, and the uncorrected estimator has nonpositive bias; demeaning does not force it to zero (\\fid{P027-E004}). The advice to total units always selected together is heuristic; for $C2$, equal sizes across pairs of clusters with zero joint inclusion probability guarantee that totalling does not increase bias (\\fid{P027-N002}).", "what_was_checked": "I checked the following, which held up.\n\\begin{itemize}\n\\item Main text: the Horvitz--Thompson estimator and its variance (p.~233); the decomposition of the variance and the definition of $A$ (p.~234); Proposition 1 and its proof, also by exact enumeration of small designs with positive and with negative bias; Proposition 2 and its proof; Corollaries 1, 2 and 3 (the cross-reference in the proof of Corollary 2 is corrected in \\fid{P027-E006}); the statements of Section 4 that $A^*\\ge A$ and that the guarantees are bounds in expectation.\n\\item Supplement, Section B: the first-order and distinct-unit joint inclusion probabilities of the one-unit-per-stratum design, the collapsed-stratum formula, and the identity $\\widehat{\\operatorname{Var}}_{C2}=2\\,\\widehat{\\operatorname{Var}}$ in that design. The rows of Table A1 agree with closed-form expectations up to the error of averaging over generated populations.\n\\item Supplement, Section C: the with-replacement estimator formula. Independent reconstructions using newly generated populations give broadly similar values for the ``True'', Horvitz--Thompson and with-replacement columns of Table A2, and the ``True'' and with-replacement columns of Table A3. These use independent Gaussian errors, sample standard deviations for the size variable, and rejection of infeasible size vectors in the random-size case; they do not recover the original simulations or establish their Monte Carlo uncertainty. The other two estimator columns of Table A3 are the subject of \\fid{P027-N004} and \\fid{P027-N005}.\n\\item Tables A1 to A3: the relative and nominal bias columns are consistent with the means and true variances within two-decimal rounding. Figures A1 to A3 visually agree with the table trends and visible points. The qualifications concern the $\\rho_{ICC}$ labels of Figure A1 (\\fid{P027-A001}), the plotting range of Figure A2 (\\fid{P027-E008}), and the legend and original-unit $C2$ series of Figure A3 (\\fid{P027-E007}, \\fid{P027-N005}).\n\\end{itemize}\nI did not check the R code for the simulations: no public copy was found (the supplement says it is available from the authors, and no request was sent), and the generated populations and seeds are likewise unavailable. No simulation was regenerated from the original inputs, and no figure was redrawn. I did not inspect the R package \\texttt{sampling} (version 2.4); access to the cited books was limited to catalogue entries, front matter and a publisher's preview. The paper's attributions to Särndal, Swensson and Wretman (1992), Wolter (2007) and Hansen, Hurwitz and Madow (1953) are neither confirmed nor contradicted. Where I list no correction, I have not established that the text is correct.\n\nFor the quadratic $C2$ correction, the exact examples of \\fid{P027-N002} show that totalling can increase bias. They do not refute the printed heuristic tendency; equal sizes for every pair of clusters with zero joint inclusion probability guarantee that totalling does not increase design bias for $C2$.", "work": { "arxiv_id": null, "audit_complete": true, "authors": [ "P. M. Aronow", "Cyrus Samii" ], "category": "Articles - Theory and Methodology", "citation": "P. M. Aronow and Cyrus Samii (2013). Conservative Variance Estimation for Sampling Designs with Zero Pairwise Inclusion Probabilities. Survey Methodology 39(1):231–241.", "doi": null, "edition": { "pages_in_audited_file": 13, "sha256": "bd92dbf0135403d1d56331fe0e52fc3f5c012e5f667abeac76fcb5b9c44dd241", "source_url": "https://www150.statcan.gc.ca/n1/pub/12-001-x/2013001/article/11831-eng.pdf", "venue": "Survey Methodology", "year": "2013" }, "id": "P027", "later_materials": null, "title": "Conservative Variance Estimation for Sampling Designs with Zero Pairwise Inclusion Probabilities", "type": "Journal article", "url": "https://pmaronow.github.io/llm-audit/works/P027/", "venue": "Survey Methodology", "year": "2013" } }